# DEEP SUMMARY - The Music of the Primes - Marcus du Sautoy

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Here is a summary of the key points from the introduction:

David Hilbert presented 23 problems in 1900 that he wanted to guide mathematics progress in the 20th century. Most were solved but the Riemann Hypothesis remained unproven.

The Riemann Hypothesis is considered the most important unsolved problem in mathematics. Hilbert said he would ask if it had been solved if waking up 500 years later.

In 1997, an announcement was posted that it had been proven. Enrico Bombieri, a renowned mathematician, sent an email about developments in a lecture by Alain Connes.

Connes is a revolutionary mathematician who developed a new geometry language. He believed it could solve major problems like the Riemann Hypothesis.

However, according to Bombieri's email, a young physicist in Connes's audience had an insight using concepts from "super-symmetric fermionic—bosonic systems" and solved it in 6 days with a computer language called MISPAR.

While a surprise a physicist solved it, mathematics and physics have become more intertwined in recent decades. The Riemann Hypothesis showed unexpected connections to physics problems.

Here is a summary of the key points about prime numbers from the passage:

Prime numbers are the basic building blocks of arithmetic, analogous to atoms in chemistry. They arenumbers greater than 1 that are only divisible by 1 and themselves.

The distribution and patterns of prime numbers have long puzzled mathematicians. There seems to be no formula to predict where the next prime will occur in the sequence of numbers.

The Riemann Hypothesis, one of the most important unsolved problems in mathematics, seeks to understand fundamental properties of prime numbers and their distribution. Proving it would have major implications.

Prime numbers have a mysterious, apparently random quality that seems at odds with mathematicians' pursuit of order and patterns. Yet they also have a timeless, universal character that exists independently of human perception.

Prime numbers have been used by twins with autism as a communication code, and were featured in Carl Sagan's novel Contact as a form of signallng between intelligent civilizations due to their universal recognition.

So in summary, it discusses the key properties and puzzles around prime numbers, their fundamental role in mathematics, and some historical contexts and examples involving primes.

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The twins John and Michael exhibited natural abilities to intuit prime numbers, similar to how autistic savants can identify dates. Doctors separated them at age 37 due to concerns their private numerical language hindered development.

Mathematicians have struggled for centuries to understand the patterns and distribution of prime numbers. In the 19th century, Riemann made breakthroughs viewing them differently and predicted an "inner harmony", known as the Riemann Hypothesis.

Proving the Riemann Hypothesis would explain the apparent randomness of primes and have huge implications across mathematics. It has become a major unsolved challenge that all mathematicians feel they must confront.

In the 1970s, the invention of RSA encryption linked prime numbers directly to security and commerce on the internet. The security of digital transactions now depends on the difficulty of factoring large prime numbers.

A solution to the Riemann Hypothesis could help generate even larger primes for improved security, or possibly help break existing codes. Businesses now have a strong financial incentive to resolve longstanding questions about prime numbers and their distribution.

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Mathematicians had been gullible to believe Bombieri's claim that he solved the Riemann Hypothesis because they were eager to see big problems solved after Fermat's Last Theorem was proven. They wanted to revel in the excitement of a major discovery.

The Riemann Hypothesis had been an unsolved problem for over a century since Hilbert included it in his famous list of problems in 1900. Mathematicians were getting more willing to take on this notoriously difficult challenge as the new century approached.

In 2000, the Clay Mathematics Institute formalized seven famous unsolved mathematics problems, including the Riemann Hypothesis, and offered $1 million prizes for solving each to incentivize progress. While not the first monetary prizes for mathematics, they were substantial and aimed to inspire more work on these enduring puzzles.

Mathematicians are motivated more by the intellectual challenge and beauty of mathematics than financial rewards. But Clay believed the million dollar prizes could attract more talent to attack these deep problems, as the Wolfskehl Prize had for Fermat's Last Theorem.

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In 1801, the discovery of the planet Ceres sparked excitement, but it soon disappeared from view without enough data to predict its trajectory.

Carl Friedrich Gauss, a young German mathematician, predicted where Ceres would reappear using a new mathematical method he had developed. His accurate prediction made him famous.

Gauss had a passion for finding patterns in numbers. Even as a child, he showed mathematical genius. He made numerous discoveries early in his career.

One of his greatest contributions was the "clock calculator" concept, which allowed arithmetic on very large numbers in a modular fashion. This opened up new patterns in number theory.

Using clock arithmetic with different numbers of "hours", Gauss revolutionized mathematics and helped uncover hidden properties of numbers that had eluded earlier mathematicians.

Gauss's innovations with modular arithmetic and clock calculators laid the groundwork for applications like modern Internet security and remain hugely influential in number theory. His work highlighted mathematics' power to predict and explain the world.

Here is a summary of the key points about primes and the universe from the passage:

Primes are numbers that can only be divided by 1 and themselves. They are the "building blocks" of all other numbers.

The ancient Greeks were the first to realize primes' importance as the fundamental units that all other numbers are built from via multiplication.

Eratosthenes, the librarian at the Great Library of Alexandria, developed one of the first systematic methods for identifying prime numbers, called the Sieve of Eratosthenes.

Carl Friedrich Gauss was fascinated by prime numbers as a child but could see no pattern or rule governing their distribution. Understanding primes was a major goal and challenge for mathematicians.

Primes seem to be randomly distributed with no obvious formula to predict the next one. Their behavior is chaotic and unpredictable, unlike things governed by known physical laws like planetary orbits.

Gauss aimed to find patterns and rules underlying nature through mathematics. The seemingly lawlessdistribution of primes was unacceptable and he sought to better understand and predict their properties.

So in summary, it outlines the historical understanding and study of prime numbers from ancient Greeks to Gauss, emphasizing their fundamental role and the ongoing challenge of finding order and patterns in their chaotic distribution.

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Fibonacci came up with a sequence of numbers based on a rabbit breeding model. This sequence appears in natural patterns like flower petals and pinecone spirals.

The Nth Fibonacci number can be expressed through a formula using the golden ratio, a special number around 1.618. Like pi, the golden ratio has an infinite non-repeating decimal expansion but appears in proportions viewed as aesthetically pleasing.

Prime numbers also arise in nature, seen in the 13- and 17-year life cycles of different cicada species. Choosing prime cycles ensures the species don't synchronize and compete too often.

Mathematicians sought a way to systematically find prime numbers. Gauss was able to find a formula for the triangular numbers through proof, demonstrating it would always work instead of testing every number.

Mathematical discovery starts with a guess or hypothesis, which becomes a theorem only once proven. Fermat's theorems were hypotheses until proven by others like Wiles. Gauss's proof of his formula moved it from a guess to a theorem.

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The test for primality fails for 341, as 341 is incorrectly declared prime when it's actually the composite number 11 * 31. This error wasn't caught until Gauss used his clock calculator with 341 hours to simplify analyzing large numbers like 2341.

Hardy described mathematical discovery and proof as mapping and exploring distant landscapes. A proof charts the path from known facts to a new discovery. It allows others to experience the same realization, not just verify each step.

Mathematicians require proof rather than just experimental evidence, as appearances can be deceiving with numbers. Prime numbers in particular take a long time to reveal patterns. Proof provides certainty that facts won't change with new discoveries.

The permenance of mathematical proof through deductive reasoning is what draws people to the field and makes mathematics unique compared to other sciences where models can change. Proofs allow establishing certainty that facts about numbers will remain true.

Mathematicians value pursuing proofs even when convinced of results, as it leads to new discoveries and the satisfaction of navigating uncharted landscapes through deductive reasoning. The proof process itself is rewarding.

Here are the key points regarding whether mathematics is an act of creation or discovery:

Many mathematicians see it as both - they feel creative when developing new ideas and proofs, but also feel they are discovering immutable truths about an external mathematical reality.

Mathematical ideas appear personal and dependent on the minds that conceive them. Yet they also believe in an objective mathematical world containing truths waiting to be uncovered.

G.H. Hardy viewed mathematics as both observations of an external reality and our own creative notes/observations of it. Others see it as more of an art under logical constraints.

Proofs can be narrated in different ways and would seem mysterious to aliens, suggesting a degree of human creativity. However, mathematicians are bound by logical steps in proofs.

Mathematics reflects both timeless truths and cultural/historical influences in how specific areas are explored. Different cultures play the "musical notes" of math concepts in their own ways.

The story of prime numbers shows both discovery of their properties but also how different eras approached them based on contemporary influences and attitudes.

So in summary, most mathematicians see it encompassing both discovery and creation - there are objective truths but also an element of human ingenuity and perspective in developing the field. It has aspects of both a scientific search and an artistic endeavor.

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Euclid proved over 2000 years ago that there are infinitely many prime numbers, but he had no way of explicitly determining primes. This left open the challenge of finding patterns in primes.

Many mathematicians like Fermat and Mersenne speculated on primes and found some intriguing properties, but fell short of providing proofs. Fermat thought his formula would always yield primes but was wrong. Mersenne correctly identified some prime numbers his formula produced.

Euler in the 18th century was able to provide explanations and proofs for many of the patterns Fermat and Mersenne discovered. His methods advanced theoretical understanding of primes.

Mathematicians still strive to improve on Euclid's insight and resolve open questions like whether there are infinitely many twin primes. Formulas have been proposed but none generate all primes and proving conjectures remains challenging given the unpredictable nature of primes. The quest to understand their patterns continues.

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Euler took 7 weeks to travel from Basel to St. Petersburg in 1727, where he pursued his mathematical dreams. His extensive output continued being published by the St. Petersburg Academy for 50 years after his death in 1783.

Catherine the Great hosted the French philosopher Denis Diderot, who disparaged mathematics. When Diderot tried to undermine the faith of Catherine's courtiers, Euler was called upon to silence him. Euler addressed Diderot with a mathematical proof of God's existence like "(a + bn)/n = x, hence God exists; reply."

Catherine was more interested in Euler's work on hydraulics, shipbuilding, and ballistics. He worked widely on mathematics including theory of music.

Euler solved the Problem of the Bridges of Königsberg by proving it was impossible to traverse the bridges only once. This established the field of topology.

Euler had a passion for numbers and prime numbers. He corresponded with Christian Goldbach sharing discoveries and attempted proofs. Euler loved calculating and produced extensive prime number tables up to 100,000.

Euler discovered the formula x^2 + x + 41 generated many primes but it could not generate all primes. Finding a simple prime-number generating formula eluded even Euler.

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Gauss was fascinated by tables of prime numbers as a teenager, even though they seemed random and unpredictable compared to logarithm tables.

Rather than trying to predict specific prime numbers, Gauss wondered if he could estimate how many primes there would be up to a given number N.

Analyzing this pattern, he discovered that the proportion of primes appeared to increase regularly based on the logarithm of N. Specifically, the number of primes up to N was roughly equal to N divided by the logarithm of N.

This linked primes to logarithms in an unexpected way. Gauss realized other bases besides 10 could be used, and primes correlated best with the natural logarithm base of e.

Gauss's insight represented a breakthrough, as earlier mathematicians focused on individual primes rather than their overall distribution. However, he provided no proof and told no one about his discovery.

Gauss prioritized rigorous proof over speculation. Without proof, he considered the connection between primes and logarithms mathematically worthless, though it pointed to deeper relationships. His reluctance to announce unproven results changed the nature of mathematics.

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For Gauss, mathematics was a private pursuit and he even encrypted some diary entries using his own secret code. Some entries like discovering that every number can be written as the sum of three triangular numbers are decipherable, but others like "Vicimus GEGAN" remain a mystery.

Some criticize Gauss for not openly sharing his discoveries, which may have held back mathematical development. However, others believe he kept results private after his treatise on number theory was rejected.

In Paris, mathematics was becoming more focused on practical applications to serve industry and war. However, prominent mathematicians like Legendre still pursued pure mathematics.

Legendre independently discovered the connection between primes and logarithms around 1798, though Gauss had achieved this as a child. Their dispute over priority was bitter.

Legendre published a prime counting function that was a better approximation than Gauss's initial formula. However, it involved an "ugly" correction term.

Gauss later refined his own more "aesthetic" function, the logarithmic integral Li(N), which proved stunningly accurate even for large numbers based on his extensive prime tables.

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Gauss had conjectured that the prime counting function Li(N) would always overestimate the true number of primes π(N), based on data up to millions. But this was eventually proved wrong - π(N) must sometime overtake Li(N), though it hasn't been observed yet due to limitations in counting far enough.

Gauss uncovered evidence that the distribution of primes follows a weighted coin toss model, with probability of 1/log(N) of a number being prime. But he did not find a way to precisely predict the outcomes.

Gaussian's conjecture, now known as the prime number theorem, aimed to prove that the percentage error between Li(N) and π(N) gets smaller as you count further. But he was distracted from a proof by his work in astronomy.

Educational reforms in Germany, led by Wilhelm von Humboldt, emphasized mathematics pursuit for its own sake rather than applications. This created an environment for more abstract mathematical thinking.

One influenced student was Bernhard Riemann, who attended the Gymnasium Johanneum school in Lüneberg. He struggled socially but worked hard, laying the foundations for his revolutionary work on prime numbers.

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Bernhard Riemann was a young, mathematically gifted boy who was encouraged and nurtured by his teacher Schmalfuss. Schmalfuss allowed Riemann access to his library where he could explore mathematics freely and find comfort in the logical world of numbers and proofs.

Riemann was drawn to conceptual, classical Greek mathematics rather than formulas. He deeply studied Legendre's book on number theory and amazingly was able to recite it perfectly after just six days.

Riemann's father wanted him to be a pastor, so he enrolled at the University of Göttingen to study theology. However, he was inspired by Gauss and the scientific tradition there, and switched to mathematics with his father's approval.

After exhausting Göttingen's resources, Riemann moved to the more intellectually vibrant University of Berlin to study under Dirichlet. Berlin had a climate that thrived on new ideas from France, unlike isolationist Göttingen.

In Berlin, Riemann immersed himself in the latest mathematical works and attended Dirichlet's inspiring lectures. He also participated in the 1848 revolution that swept through the city. This exposure to new ideas from France and interactions with other mathematicians helped propel Riemann's own breakthroughs.

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Augustin-Louis Cauchy was a French mathematician who lived in the late 18th/early 19th century. He came of age shortly after the French Revolution.

As a young boy, he preferred intellectual pursuits over physical exercise due to limited food availability during that period. The established field of mathematics provided a refuge for him.

The mathematician Lagrange recognized Cauchy's talent and advised his father not to let him study mathematics until age 17, to develop his broader literary skills first.

This advice proved sound - when Cauchy finally studied mathematics intensely, he developed a new, innovative style of writing and thinking about mathematics.

However, his abstract, pure mathematical approach was criticized by some contemporaries and school directors who felt it detracted from more practical applications of math.

Cauchy's works greatly influenced later mathematicians like Bernhard Riemann. Riemann immersed himself in Cauchy's outputs and emerged declaring it revealed a "new mathematics."

This "new mathematics" involved imaginary numbers, which seem contradictory but provide flexibility and solutions to equations. Cauchy and Riemann helped establish imaginary numbers as a legitimate part of mathematics.

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In the early 19th century, mathematicians like Gauss were beginning to represent imaginary numbers as points on a map, where moving east represented multiplying by i and north represented multiplying by 1. However, Gauss kept his "map of the imaginary world" secret, knowing pictures were viewed with suspicion at the time.

Mathematicians were wary of pictures following misleading proofs based on intuitive images. They sought rigorous proofs using formulas alone.

In the late 18th century, Euler began exploring functions that took imaginary numbers as inputs rather than just real numbers. This led to unexpected discoveries, like the sine function emerging from the exponential function with imaginary inputs.

Riemann was influenced by these developments when studying under Gauss in the 1840s-50s. He explored functions over imaginary numbers, building on Cauchy's work to make this rigorous. His 1851 dissertation on the topic impressed Gauss.

Riemann struggled socially in Göttingen but found support from Dirichlet and struck up a friendship with physicist Weber, Gauss's former collaborator on projects like the telegraph.

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Dirichlet was fascinated by Gauss's work on number theory, especially his conjecture related to prime numbers distributed across a clock face (known as Fermat's little theorem).

Dirichlet proved this conjecture using a mathematical function called the zeta function, defined as an infinite sum involving exponential terms and reciprocals. This was an innovative application of different areas of math.

The zeta function originated from interest in the mathematical relationship between music and harmonies discovered by Pythagoras. Euler also studied this connection.

Many mathematicians are naturally drawn to both mathematics and music due to numerical underpinnings and aesthetic qualities shared between the disciplines.

During a walk, Dirichlet inspired Riemann's next work by discussing the zeta function. This opened up a new perspective on primes for Riemann using complex analysis and the zeta function. Dirichlet recognized Riemann's originality and they had many philosophical discussions that helped catalyze Riemann's ideas.

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Doing mathematics can feel like randomly striking notes on a piano until suddenly discovering a harmonic combination that stands out for its inner beauty. Mathematicians often take pleasure in re-examining proofs to uncover subtle nuances of how the ideas fit together harmoniously. Both mathematics and music involve technical languages of symbols used to articulate patterns.

There are also physical overlaps between mathematics and music. The harmonics that give each musical instrument its unique sound stem from mathematics. The infinite harmonic series in music relates to the zeta function and Euler's discovery that its value when fed 1 is related to pi. Euler also connected the primes to the zeta function through his product formula, providing new ways to think about prime numbers. This inspired later mathematicians like Dirichlet and Riemann to gain new insights into prime distribution using the zeta function approach. Riemann in particular believed he was close to proving Gauss's prime number conjecture through this method.

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Riemann discovered that the zeta function could provide a new perspective on prime numbers by transforming them into the zeros of the function. This allowed patterns and regularities in the distribution of primes to potentially be revealed.

Riemann published his landmark paper on the zeta function and prime numbers in 1859. While the paper contained gaps and unfinished proofs, it outlined totally new ideas and approaches that would shape the field of number theory going forward.

A key conjecture presented by Riemann, though he admitted he could not prove it, was what is now known as the Riemann Hypothesis - that all nontrivial zeros of the zeta function lie on the critical line with real part of 1/2. Solving this would carry a million dollar prize today.

Riemann used his strong geometric intuition to depict the zeta function as existing in four dimensions. He envisioned a "landscape" or "graph" that could capture its behavior when fed imaginary numbers. Though impossible to draw, this helped conceptualize its properties.

One way to understand this higher-dimensional object is through its three-dimensional "shadow." Riemann's analysis of this shadow revealed peaks and patterns in the distribution of the zeros, opening up investigations into the underlying structure of prime numbers.

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Riemann was exploring an imaginary landscape defined by the zeta function. The landscape seemed to end at the number 1, as the zeta function spirals to infinity for numbers west of 1.

However, Riemann believed the actual landscape should not end there. He found another formula that could be used to construct the missing landscape to the west, joining it seamlessly to the existing landscape east of 1.

Riemann discovered two key facts about these imaginary landscapes. Their geometry is extremely rigid, with only one way to expand the landscape. And the locations of points where the zeta function outputs zero contain enough information to reconstruct the entire landscape.

Riemann realized the zeros are like the spectrum of light from a chemical, revealing information about the whole. He found a direct connection between prime numbers and the zeros - his formula used the zeros to give an exact count of primes up to any number N, correcting errors in previous estimates.

Each zero corresponds to its own wave. Higher zeros produce faster waves. Riemann discovered the heights of these waves encode the corrections to estimates of primes up to N, giving the exact number. His work revealed primes, zeros and waves are transformed versions of each other in this imaginary landscape.

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Riemann discovered waves arising from the zeros of the zeta function that describe and control the distribution of prime numbers. Over infinite waves, the graph transforms from a smooth one to the jagged staircase pattern of primes.

These waves reveal harmonic structure and hidden musical qualities of the primes. Riemann's formula captures the subtle harmonies in a way no one had achieved before.

Fourier analyzed heat propagation and sound waves using sine curves as building blocks. He showed tones of different instruments and even white noise result from combinations of pure sine waves.

This was controversial at the time but proved sounds and other phenomena could be broken down and reconstructed from simple sine curves. Fourier applied this approach more broadly to depict mathematical and physical graphs.

His work laid the foundation for modern harmonic analysis and representations of signals through Fourier transforms, which have widespread applications from music to imaging to data analysis. Riemann and Fourier uncovered deep connections between numbers, music, physics and mathematics.

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This passage discusses Riemann's famous 1859 paper where he analyzed the distribution of prime numbers using complex analysis and the Riemann zeta function.

Riemann treated the locations of the zeros of the zeta function as musical notes that, when combined, reproduced the "sound" of the primes. Each zero corresponded to an isolated musical note.

Remarkably, the first few zeros Riemann calculated appeared to lie on a straight north-south line in the complex plane (the "critical line"). This pattern was totally unexpected and became known as the Riemann Hypothesis.

If all zeros indeed lie on this line, it would imply that Gauss's conjecture about the distribution of primes is always accurate. So proving the hypothesis was important for mathematics.

However, Riemann admitted he did not attempt a rigorous proof in his 1859 paper. He was more focused on verifying Gauss's conjecture.

The discovery of this pattern in the zeros transformed the study of prime numbers by providing a new "imaginary landscape" for mathematicians to explore. But Riemann never returned to the subject himself.

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In 1885, Thomas Stieltjes, a little-known Dutch mathematician, claimed to have proved Riemann's Hypothesis. This would also prove Gauss's famous prime number conjecture.

Stieltjes struggled in school but was talented in mathematics. He corresponded with prominent French mathematician Charles Hermite, who strongly supported him.

Hermite believed Stieltjes' claim, hoping it would earn Stieltjes great recognition. However, when pressed for details, Stieltjes was reluctant to provide a full proof.

By 1890, with no proof forthcoming, Hermite proposed a prize from the Paris Academy dedicated to proving Gauss's conjecture, believing Stieltjes could win it. Hermite felt Stieltjes only needed to prove a small part of Riemann's ideas, not the full hypothesis.

Ultimately, Stieltjes never produced a complete proof, leaving others still trying to realize Riemann's revolutionary ideas about the distribution of prime numbers. His claim raised expectations but also demonstrated how difficult proving the Riemann Hypothesis would be.

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The article discusses the mathematical journey towards proving the Riemann Hypothesis and Prime Number Theorem.

Hadamard was able to partially prove the Prime Number Theorem in 1896, showing that there were no zeros on the critical line past 1, proving Gauss's conjecture. However, he fell short of fully proving the Riemann Hypothesis.

David Hilbert helped bring widespread attention to Riemann's ideas and the goal of proving the Riemann Hypothesis. As a professor at Göttingen, the center of mathematics at the time, he helped launch efforts to fully solve this problem in the 20th century.

While making progress, mathematicians were still far from a full proof of the Riemann Hypothesis, which became seen as the great unsolved problem in mathematics. Efforts to prove it continue to this day.

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David Hilbert became fascinated with new types of geometry proposed in the 19th century that questioned one of Euclid's fundamental axioms about parallel lines. He studied these non-Euclidean geometries from an abstract, logical perspective rather than focusing on physical reality.

Earlier, Carl Gauss had also considered these alternative geometries but did not publicly discuss them out of concern for contradicting the Greeks. As a young man, Gauss speculated there could be geometries without parallel lines or with multiple parallel lines through a point. He later conducted experiments hoping to observe non-Euclidean effects from light bending over long distances, but the measurements were too small.

In the 1830s, Nikolai Lobachevsky and Janos Bolyai publicly proposed the first non-Euclidean geometries. However, they were initially dismissed. Hilbert began exploring the logical foundations and relationships between these geometries and Euclid's. He realized no one had proven Euclid's geometry was logically consistent on its own.

This led Hilbert to question the foundations of mathematics more broadly. In 1900, he gave a seminal lecture at the International Congress of Mathematicians outlining 23 major open problems in mathematics. The lecture helped launch a new era focusing on rigorous foundations and abstract theory in mathematics.

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Hilbert gave a lecture at the 1900 International Congress of Mathematicians to deliver his famed list of 23 problems and challenge mathematicians to solve them in the new century. He hoped to sweep away pessimism about limits to understanding nature and inspire workers in the field. Some problems concerned foundational questions, while others ranged widely across mathematics. Notably, the 8th problem specifically posed proving the Riemann Hypothesis, which Hilbert believed was the most important problem in mathematics. He thought solving it and fully understanding Riemann's prime number formula would illuminate many mysteries. Hilbert aimed to redirect focus onto abstraction rather than any one problem. Although his lecture faced oppressive weather, it cemented Hilbert's reputation as a pioneer and ensured the problems would hugely influence 20th century mathematics.

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Landau was a renowned mathematician who helped establish Göttingen as a center of mathematics in the early 20th century. He had high standards and demanded rigorous proofreading from his students.

Landau and Harald Bohr made early progress on the Riemann Hypothesis by showing that most zeros of the zeta function are clustered close to Riemann's critical line. However, they could not prove that most lie exactly on the line.

G.H. Hardy later proved that infinitely many zeros do lie on the critical line, providing the first proof that some zeros satisfy the hypothesis. However, his result did not prove that all or even half of the zeros lie on the line.

While an important achievement, Hardy's work only accounted for an infinite but still proportional zero number of zeros. Fully proving the Riemann Hypothesis remained an tantalizing open problem that obsessed both Landau, Bohr, Hardy and others like Hilbert. Significant progress had been made but a complete proof eluded even the field's best mathematicians.

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G.H. Hardy had a profound interest in prime numbers from a young age and worked to communicate the beauty and intrigue of number theory to broader audiences.

He had a lifelong battle against the concept of God, trying to disprove God's existence. He took unconventional steps like carrying extra sweaters on trips to try and trick God.

He visited Harald Bohr regularly and they would try daily to prove the Riemann Hypothesis without success. One time on a boat trip back from Denmark, Hardy jokingly sent a postcard saying he solved it as an "insurance policy" against God letting the boat sink.

Hardy's passion for the Riemann Hypothesis helped elevate it within mathematics. While praising beauty in proofs, Hardy's own proofs often relied on extensive technical details.

He collaborated very successfully with J.E. Littlewood for over 30 years, combining Hardy's elegance with Littlewood's forceful problem-solving style. They established clear "axioms" for their collaboration.

So in summary, Hardy had an outsized impact on number theory through his dedication to communicating its delights, solving hard problems, and lengthy collaboration with Littlewood. He remained deeply eccentric in his philosophy against God.

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J.E. Littlewood had an outstanding career as a mathematician after excelling as an undergraduate at Cambridge, earning the title of senior wrangler.

As a new graduate, his tutor Ernest Barnes gave him the problem of investigating the zeros of the zeta function to work on over the summer, unaware of how difficult this problem (the Riemann Hypothesis) actually was.

While Littlewood made no progress on proving the hypothesis, he did rediscover the connection between the zeta function and prime numbers. Although this was already known on the continent, it was still novel in Britain at the time.

G.H. Hardy recognized Littlewood's potential and they began their famous collaboration once Littlewood joined Trinity College, Cambridge in 1910. They were inspired by Landau's recent work connecting primes and the zeta function.

One of Littlewood's great early contributions was disproving Gauss' second conjecture about his logarithmic integral function always overestimating the number of primes up to a given value. This was a major achievement to disprove a hypothesis Gauss believed to be true.

Littlewood also showed that Riemann's refinement of Gauss' conjecture would not always be more accurate than the original, challenging other prevailing beliefs at the time.

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Littlewood's proof had a huge effect on how mathematicians viewed number theory and primes. It showed that primes can only be truly understood through rigorous proof, not empirical evidence or vast calculations.

Mathematicians began building more of their work on the assumption that the Riemann Hypothesis is true, as it helped progress problems. However, there was always a risk that it could be disproved later.

Meanwhile in India, the clerk Srinivasa Ramanujan became obsessed with studying primes in his spare time with no formal training. He filled notebooks with observations and calculations.

Ramanujan claimed his insights came to him in dreams sent by the goddess Namagiri. His intuition and naivety outside Western modes of thought allowed him to make new discoveries.

Mathematicians like Hadamard believed creativity involved an incubation period where the subconscious was free to explore ideas seeded in the conscious mind. Ramanujan seemed able to achieve this dreamlike state while awake.

So in summary, it contrasts the developments in rigorous proof-based number theory with Ramanujan's intuitive, dream-inspired approach to making discoveries about primes with no formal education.

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Ramanujan failed his college exams in 1907 due to being required to study subjects like English, history, and physiology that were irrelevant to his interests in mathematics.

By 1910, he was publishing some of his ideas in journals and his name came to the attention of mathematicians in India. C.L.T. Griffith recognized his work was remarkable but couldn't fully understand it.

Griffith sent Ramanujan's papers to Professor Hill in London, who dismissed most of it as meaningless. However, Hill was not totally dismissive and encouraged Ramanujan.

Ramanujan sent letters with his ideas to mathematicians in Cambridge, including G.H. Hardy. Hardy was initially bored and irritated by the letters which contained wild theorems but no proofs.

After further examination with J.E. Littlewood, they realized Ramanujan was a genius, as he had reproduced major discoveries by Riemann without formal training.

Hardy eagerly sought more details and proofs from Ramanujan. Further correspondence showed Ramanujan had independently discovered other fundamental results, though proofs were still lacking.

Hardy and Littlewood recognized Ramanujan's extraordinary natural talents and insights into difficult mathematical problems, despite his lack of formal proof abilities. They were excited to nurture his potential with their guidance and expertise.

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Ramanujan claimed in a letter to Hardy that he had found a precise formula to reconstruct Riemann's zeta function and count primes accurately.

Littlewood proved while on vacation that Ramanujan's claim was incorrect. The errors from the zeta function's zeros could not cancel each other out perfectly, as Ramanujan suggested. There would always be some error, no matter how far the counting went.

However, Littlewood's analysis did provide a new insight - it confirmed that the Riemann Hypothesis predicted the smallest possible error in counting primes up to a number N.

Ramanujan was under the misconception that the zeta function had no points at sea level, which would have made his formulas accurate if true. Littlewood was still impressed by Ramanujan's ability.

Hardy and Littlewood worked to bring Ramanujan up to date on current mathematical knowledge and techniques, as he was largely self-taught. The transition to Cambridge was also a big culture shock for Ramanujan.

However, the collaboration was extremely fruitful, with Ramanujan producing many new theorems on an almost daily basis. They made progress on problems like Goldbach's Conjecture.

While Ramanujan's attempts at precise prime counting formulas were incorrect, his ideas in related areas like partition numbers were more successful and influential.

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Ramanujan succeeded in finding an exact formula for partition numbers, which count the ways of dividing integers into piles, but failed to find a similar formula for prime numbers.

Working with Hardy, their formula for partition numbers involved complex functions like square roots, pi, differentials, and trigonometric functions. It produced answers that were correct when rounded to the nearest whole number.

Their collaboration initiated the Hardy-Littlewood circle method, which they later used with some success to make progress on Goldbach's conjecture about sums of primes.

Ramanujan's health declined in England due to malnutrition, homesickness, and the harsh climate. He tried to commit suicide and spent time in sanatoriums before eventually returning to India, where he died at age 33.

Although he didn't solve the prime numbers problem, Ramanujan left many insights that continued fueling mathematicians for decades. Some of his conjectures, like the tau conjecture, were only fully understood and solved much later.

His notebooks still hold mysteries, like apparently accurate counts of primes below 100 million that can't be explained by his known work. Scholars wonder if he was onto a prime formula we have yet to discover.

Here is a summary:

In 1976, a lost notebook by Ramanujan containing new mathematics was discovered, increasing speculation that more of his work remains undiscovered.

Ramanujan's death in 1920 came as a shock to Hardy. Hardy considered their collaboration one of the highlights of his career, which gave him consolation in his later life.

As Hardy aged, he struggled with depression and losing his abilities in mathematics. In his book "A Mathematician's Apology", he described the challenges of being an older mathematician.

Like Ramanujan, Hardy attempted suicide by overdosing on pills, though he vomited them up. C.P. Snow recalled visiting the sick Hardy, who was self-mocking about his failed suicide attempt.

In the 1920s-30s, some mathematicians like Landau, Hardy, and Littlewood grew skeptical of Riemann's evidence for the Riemann Hypothesis. Though his 1859 paper was groundbreaking, it lacked calculations validating the locations of zeros.

Hardy and Littlewood developed a method to calculate some early zeros, hoping to disprove the hypothesis. By 1929, applying their method to 138 zeros, all were found to align with Riemann's prediction.

Here is a summary:

Hardy and Littlewood had developed a formula for calculating the locations of zeros of the zeta function, but it was becoming computationally infeasible to use to find zeros farther north (with higher real part values). Mathematicians were giving up on explicitly locating zeros.

Siegel discovered unpublished notes of Riemann's (known as his Nachlass) that had been saved by Riemann's family and stored in a library.

Upon examining the notes, Siegel realized Riemann had developed a more powerful formula for calculating zeros than what was currently known. Riemann had anticipated parts of Hardy and Littlewood's work and also discovered new methods.

Riemann's formula allowed for more accurate calculations farther north than previous methods. It confirmed over 1,000 zeros lay on the critical line, as the Riemann Hypothesis predicted.

The discovery changed perceptions of Riemann from a theorist to also being a skilled calculator and experimenter empirically studying the zeta function.

Riemann may have recorded further thoughts in an unpublished "little black book" from 1860 that has been lost. Its contents could provide new insights if ever found.

Here is a summary:

The passage describes the difficult environment for mathematicians in Germany during the rise of the Nazis in the 1930s. Many Jewish and left-leaning mathematicians lost their jobs and had to flee Germany.

Even prominent non-Jewish mathematicians like Landau faced harassment from Nazi students. He eventually resigned and retired to Berlin, unable to continue teaching in Göttingen.

By the late 1930s, Göttingen's prestigious mathematics department had been largely destroyed by Hitler's purge of the universities. Many mathematicians emigrated to the US to escape.

During this period, the isolated mathematician Atle Selberg in Norway was able to intensely focus on his work, cut off from developments elsewhere due to the war. He made breakthroughs on partition numbers, improving on Ramanujan and Rademacher's formulas.

When Siegel met up with Harald Bohr after the war, Bohr surprised him by mentioning Selberg's accomplishments during the years of Siegel's self-imposed exile in America to protest the war in Germany.

Here is a summary:

Atal Bosworth Hardy and John Littlewood had proven there were an infinite number of zeros of the Riemann zeta function on the critical line, but could not show this accounted for even a fraction of the total zeros.

Atle Selberg believed there was more that could be done with their methods. He developed new techniques and showed that the percentage of zeros on the critical line did not decrease as you moved further along the line. This was the first substantial evidence that many zeros lie on the line, in line with the Riemann Hypothesis.

Selberg's breakthrough established him as a leader in the field. He was invited to speak at a congress in Copenhagen, where it was confirmed he was well ahead of others.

Selberg was recruited to the Institute for Advanced Study in Princeton, helping establish it as a new center for mathematics and drawing top scholars from war-torn Europe.

Hungarian mathematician Paul Erdos also emigrated to the US. He was a prolific collaborator known for his eccentric habits and beliefs about mathematics originating from a "Great Book." Erdos and Selberg's careers would become intertwined, though they had differing work styles.

Here is a summary:

Paul Erdos was a renowned Hungarian mathematician passionate about prime numbers from a young age. His father showed him proofs that stimulated this interest, like Euclid's proof that there are infinitely many prime numbers.

Erdos was fascinated by problems like predicting the longest stretches of numbers without any primes, and how long you need to count from a number before finding the next prime.

In 1845, Joseph Bertrand conjectured that counting from any number N to 2N, you are guaranteed to find a prime. This became known as Bertrand's Postulate.

Within seven years, Pafnuty Chebyshev proved Bertrand's Postulate using ideas from his own work establishing bounds for the prime number theorem.

As a teenager, Erdos proved Bertrand's Postulate but was disappointed to find Ramanujan had already simplified the proof. This led Erdos to explore how large gaps between primes could be.

At the Institute for Advanced Study, Erdos proved a conjecture of Mark Kac about the randomness of prime divisibility of numbers, sparking Erdos' passion for combinining number theory and probability.

The order of zeros in the Riemann zeta function was later understood to explain the apparent randomness of primes while still allowing for predictable properties like bounds on counting primes.

Here is a summary:

Selberg discovered an elementary proof of Dirichlet's theorem about primes in certain residue classes, avoiding sophisticated tools like the zeta function.

Erdos saw that Selberg had also proved a formula useful for improving Bertrand's postulate on primes between N and 2N. Erdos used this formula to prove primes exist between N and 1.01N for large N.

Selberg realized this result, combined with his other work, gave a full elementary proof of the prime number theorem, which had previously required tools like the zeta function.

When Selberg presented the work, Erdos invited a large audience without Selberg's knowledge. Erdos then took credit for the key intermediate result, though Selberg had discovered it.

Selberg wanted to publish separate papers due to never collaborating before, while Erdos insisted on a joint paper in the style of Hardy and Littlewood. This led to a disagreement over credit for the elementary proof of the prime number theorem.

The story shows differing temperaments, with the more solitary Selberg preferring to work alone versus Erdos' highly collaborative approach to mathematics. It caused controversy over who deserved most credit for the breakthrough.

Here is a summary of the key points:

Amadeus Selberg and Paul Erdos had a famous feud over credit for an elementary proof of the Prime Number Theorem. Selberg published the proof alone after Erdos offered a step, going against mathematical convention of collaborating with Erdos.

Erdos was furious at not being credited and appealed to Hermann Weyl, who sided with Selberg. Their conflict highlighted the importance of credit and priority in mathematics.

Selberg's proof was not as breakthrough as hoped but established a method. Later mathematicians like Norman Levinson improved on Selberg's estimates of the percentage of zeros of the Riemann zeta function that lay on the critical line, getting as high as 98.6% before a mistake.

The current record is Brian Conrey's 1987 proof that 40% of zeros lie on the critical line. Proving more than 50% is seen as important but very difficult.

Selberg still believes the Riemann Hypothesis will be proved but the proof may be too complex for the human mind to follow. Evidence strongly supports it being true.

Alan Turing's work at Bletchley Park on codebreaking machines during WWII laid the foundations for later development of computers, which could generate new evidence about the Riemann zeta function and its zeros.

Here is a summary:

Turing was fascinated by machines and their potential applications in mathematics from an early age. At Cambridge in the 1930s, he sought to use machines to tackle two of Hilbert's famous problems.

The first was Hilbert's second problem of proving mathematics is logically consistent. Turing devised a "theoretical machine" that would undermine efforts to establish a secure foundation for mathematics.

The second was Hilbert's eighth problem of proving or disproving the Riemann Hypothesis. Turing believed physical machines could be built to search for zeros off Riemann's critical line, which could disprove the hypothesis.

-Meanwhile, Gödel had stunned the mathematical world in 1930 with his incompleteness theorems. He proved mathematics cannot prove its own consistency, dashing Hilbert's program. Mathematical foundations could never be fully secured.

-Turing was aware of this crisis in foundations sweeping through Cambridge. Despite his isolation, he sought to use real and theoretical machines to tackle problems in mathematics that colleagues were failing to resolve through traditional means.

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Gödel's incompleteness theorems proved that within any consistent axiomatic system, there will always be true statements that cannot be proved. Math was built on an infinite tower of turtles with no final foundation.

This contradicted Hilbert's belief that all mathematical truths could ultimately be proved. It showed proof has inherent limitations and math contains an irreducible element of uncertainty.

Gödel used prime numbers and Gödel numbering to demonstrate unprovable but true statements. This challenged the view that proof was the sole path to mathematical truth.

His results shook mathematicians and raised doubts about proving conjectures like Goldbach's, the twin prime conjecture, and the Riemann hypothesis within current foundations.

However, Gödel did not undermine proven theorems. His work showed math's reality extends beyond deduction from axioms. Foundations must evolve alongside continued construction.

This opened questions about a "decision procedure" that could determine provability without finding proofs. Turing was fascinated by this and began considering a "mechanical" procedure or "oracle", anticipating the concept of a computing machine.

So in summary, Gödel introduced foundational uncertainty in math and influenced Turing's early thinking about mechanical proofs/procedures, challenging views of both proof and the nature of mathematical truth and provability.

Here is a summary of the key points:

Turing was fascinated by Gödel's Incompleteness Theorem as explained by Newman in lectures at Cambridge in the 1930s. This sparked Turing's interest in Hilbert's "Decision Problem" - could one construct a machine to determine if mathematical statements are provable or not.

Turing had the insight that "Turing machines" - hypothetical machines that could model any logical mathematical process - were not powerful enough to solve the Decision Problem.

He drew on Georg Cantor's work demonstrating different sizes of infinities to argue that, like Gödel's theorem, any machine attempting the problem would miss statements, just as fractions don't cover all real numbers.

Turing worked out the details of the argument with Newman over a year, concerned about objections.

Unfortunately, Alonzo Church at Princeton had arrived at the same conclusion around the same time but published first.

However, Turing's concept of a "universal machine" that could simulate any other was more practical and influential, effectively founding the concept of a general-purpose computer.

This theoretical work was a foundational contribution to proving limitations of logic and establish computability theory, even if not fully recognized at the time. It proved highly significant for later computer development.

Here is a summary:

Turing arrived in Princeton in 1937 but found that Gödel had returned to Austria. He did meet G.H. Hardy, who was visiting, though Hardy was initially standoffish.

After proving Hilbert's decision problem, Turing looked for another big problem and thought of the Riemann hypothesis. He got papers on it from colleague Albert Ingham and discussed it with Hardy.

By 1937 Hardy was more pessimistic about proving the hypothesis and thought it may be false. Turing wanted to build a machine to disprove Riemann.

Inspired by machines that calculate tidal patterns via periodic functions, Turing proposed building a "zeta machine" to calculate the Riemann zeta function landscape and potentially find a point off Riemann's critical line.

The idea drew on Babbage's 19th century proposal to build machines to compute mathematical tables via gears. Turing began constructing his zeta machine but WW2 interrupted the project.

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After WWII, Turing began working to build a universal computing machine that could be programmed to perform different tasks. He worked at the Royal Society Computing Laboratory in Manchester with Max Newman.

In Manchester, Turing used his codebreaking experience from Bletchley Park to design a program that could use their new computer to explore Riemann's Hypothesis and search for counter-examples.

Turing's machine managed to check the first 1,104 zeros of the Riemann zeta function before breaking down. However, Turing's personal life was collapsing after he was arrested and convicted for his homosexuality.

In 1954, Turing died from apparent cyanide poisoning. His mother claimed it was accidental but most believe it was suicide.

Turing's work began the era of using computers to further explore the Riemann Hypothesis, though it would be some time before "Riemann rovers" could travel much further along Riemann's line.

Turing helped establish that machines could not fully answer all mathematical questions, relating to Hilbert's programme, but Julia Robinson was investigating if machines could say anything about solutions to equations, like Hilbert's 10th problem.

So in summary, it outlines Turing's pioneering work on building a universal computer to explore mathematical problems like the Riemann Hypothesis, the collapse of his personal life, and how his work related to broader questions around computation and foundations of mathematics.

Here is a summary:

Julia Robinson was fascinated by a radio program describing how mathematicians D.N. and D.H. Lehmer built early computing machines to tackle mathematical problems. This piqued her interest in using computational approaches.

As a student, she was intrigued by Gödel and Turing's results showing limitations of logical and computational methods via paradoxes and undecidability. However, she believed in an objective mathematical reality grounded in numbers.

With her husband Raphael Robinson, she focused on Hilbert's 10th problem of providing an algorithm to determine if equations have solutions. Work by Gödel, Turing and others suggested this was not possible.

Julia Robinson hypothesized a connection between Turing machines and equations such that solving one type of problem could translate to the other. This became an obsession as she aimed to disprove the possibility of Hilbert's algorithm.

After many years of work with Martin Davis and Hilary Putnam, she narrowed the problem down to finding an equation for a specific numerical sequence. Russian mathematician Yuri Matijasevich then found this last piece, completing the solution and proving Hilbert wrong.

Robinson was thrilled to see her long-held conjecture finally proven true, showing the limitations of computational methods in mathematics. It was a landmark achievement uniting scholars across borders.

Here is a summary:

Matijasevich and Robinson both contributed significantly to solving Hilbert's 10th problem, the insolvability of Diophantine equations, but neither claimed full credit and recognized the collaborative nature of the work.

While they proved the problem in the negative sense (there is no general method to determine solutions), their work showed that any numbers produced by a Turing machine could be expressed through equations.

This meant there had to exist a formula to generate prime numbers, since primes can be produced by a Turing machine. Such a formula was eventually discovered in 1976, involving 26 variables, that could systematically generate all prime numbers through calculations.

However, the discovery of this prime-generating formula was not hailed as greatly as it may have been in the past, as mathematics had progressed beyond studying individual formulas. The formula was also practically useless and did not reveal anything uniquely special about primes compared to other number sequences.

So in summary, it outlines the key contributions to Hilbert's 10th problem and the subsequent discovery of a prime-generating formula, but notes it was not as significant an achievement as it may have been previously due to changes in mathematics.

Here is a summary:

The largest known prime number increased dramatically with the advent of computers, from 39 digits in WWII to over 1 million digits today.

Turing was interested in using early computers to find large prime numbers, starting with Mersenne primes of the form 2^n - 1 where n is prime. This was a good task for the limited memory of early computers.

Lucas and Lehmer developed an efficient test for checking if Mersenne numbers are prime by generating related Lucas-Lehmer numbers and testing divisibility.

Turing pursued finding Mersenne primes on early Manchester machines after WWII but was unsuccessful in breaking the record.

The search for largest primes was dominated by large computer systems like Cray supercomputers through the 1990s.

Recently, networked groups of ordinary desktop computers have started breaking prime number records by distributing the computation over many machines using Internet resources. This shifts the advantage from large centralized computers to distributed "ant colonies" of smaller machines.

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The search for Mersenne primes was coordinated over the internet, with amateur scientists using their computer's idle time to scan numbers for new record primes.

One participant enlisted his company's 2,585 computers to help, slowing down the network. When the FBI investigated, he admitted it was "too tempting" and was fired.

In 1996, an American programmer launched a project called GIMPS that recruited over 200,000 computers worldwide to collaborate on finding Mersenne primes.

The first internet discovery was made by a French programmer in 1996. Subsequent primes were found by participants in England, California, and Michigan (the largest at over 2 million digits, winning $50k).

Wired magazine covered GIMPS, marking a shift to computers playing a major role in prime number searches alongside traditional methods.

While computers can outperform humans in calculations, they lack the imagination and critical thinking to truly understand mathematics. Finding large primes doesn't prove theoretical conjectures like whether Mersenne primes occur infinitely often.

The first theorem proven by computer was the Four Color theorem in 1976, using 1,200 hours of computation to verify 1,500 basic maps - illustrating computers' usefulness but inability to provide understanding. The computer gave an answer but not deeper insight.

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The computer proof of the Four-Color Theorem was seen as laborious and lacking in deep understanding, rather than providing an "Aha!" moment of insight.

There was debate over whether computers could help prove the Riemann Hypothesis. If it's false, a computer could find a zero off the line. But if true, a computer could only provide increasing evidence but not a proof.

In the 1960s/70s, computers began exploring Riemann's landscape farther using a secret formula from Riemann's unpublished notes. This provided more evidence but not a proof. Some saw computing as just covering up lack of real progress.

Don Zagier was a leading skeptic of the Riemann Hypothesis. Enrico Bombieri strongly believed in it as an "article of faith". They made a bet - Zagier would believe over 300 million computed zeros on the line.

Zagier's analysis was that 300 million zeros represented a threshold where zeros should start deviating if the hypothesis was false, making it a meaningful test of the hypothesis.

Here is a summary:

Don Zagier made a bet with Enrico Bombieri that the Riemann Hypothesis is true. The threshold for the bet was calculating 300 million zeros along the critical line without finding a zero off the line.

Zagier devised a new graph method to analyze the behavior of zeros along the critical line. As more zeros were calculated, the graph appeared to be repelled from crossing the line, suggesting the hypothesis is true.

In the 1970s, computers started calculating more zeros, reaching 75 million by 1978. By 1979, a joint team calculated 200 million zeros, still all on the line.

Zagier worried they may calculate the full 300 million and force him to lose the bet. But they stopped at 200 million, to his relief. However, Hendrik Lenstra then encouraged them to keep going to 300 million to make Zagier lose, which they did.

Jan van de Lune has since confirmed the first 6.3 billion zeros obey the hypothesis. Calculation of zeros provides evidence for the hypothesis but cannot prove it.

In the 1980s, AT&T researcher Andrew Odlyzko calculated zeros and helped disprove the related Mertens Conjecture, showing calculation's ability to destroy conjectures. Odlyzko continued significant calculations supporting the Riemann Hypothesis.

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The passage required Odlyzko to know as accurately as possible the location of the first few zeros in the zeta landscape. This was because high-precision calculations are well suited for computers. When Odlyzko gained access to AT&T's new supercomputer, he used it to determine the exact location of the first million zeros. His calculations helped disprove the long-standing Mertens Conjecture, showing that even seemingly accurate numerical predictions can still be wrong. Analyzing the locations of zeros to high precision has continued to provide insights into the Riemann Hypothesis and properties of prime numbers.

Here is a summary:

Encryption systems during WWII, like the Enigma machine, relied on codebooks that contained the day's encryption settings. If an enemy gained access to the codebook, they could decrypt all messages.

Using such a system for modern internet communications would be logistically difficult. Users would need to receive secure letters informing them of each website's encryption settings before shopping or sending banking details. With high internet traffic, many of these letters would likely be intercepted.

Mathematicians like those at Bletchley Park who cracked Enigma developed new "public key cryptography" systems suited for rapid global communication. Instead of single keys, they use two different keys - one to encrypt and one to decrypt.

At MIT in the 1970s, Ron Rivest, Adi Shamir and Leonard Adleman were inspired by Diffie and Hellman's 1976 paper proposing public key cryptography. They worked to implement this idea in a practical cryptosystem.

After exploring many mathematical problems and discarding ideas, they had a breakthrough realizing factorization could provide the basis for an effective public key system. This became known as RSA encryption, one of the most important and widely used cryptosystems.

Here is a summary:

Rivest had the idea to use prime numbers to crack codes after a Seder dinner where he drank wine and realized factoring numbers into primes could be used to encrypt/decrypt messages securely.

He shared the idea with Adleman and proposed their names be on the paper describing the new public key encryption system they called RSA. Adleman insisted his name be taken off at first, thinking it wasn't that important, but later agreed to be third author.

Martin Gardner wrote about RSA in his Scientific American column, generating huge interest. Thousands wrote to Rivest asking for details. This convinced Adleman the idea could be big.

Early attempts to commercialize RSA or interest intelligence agencies were unsuccessful as the technology wasn't mature yet. But it eventually became the standard for secure Internet transactions.

RSA works by using prime number factoring properties described by Fermat and Euler centuries ago. A message is encrypted using one prime number "clock" and can only be decrypted using the product of two prime numbers, forming the basis for public/private key encryption.

So in summary, it describes the origins and initial reactions to the foundational RSA public key encryption system invented by Rivest, Adleman and Shamir based on number theory ideas from Fermat and Euler.

Here is a summary of the key points:

The clock calculator method is used to select a public key for encryption. It picks two large prime numbers p and q and multiplies them to get a very large number N that determines the "hours" on the clock face.

Customers get an encoding number E to encrypt their credit card number C by raising it to the power of E on the clock calculator (modulo N).

While the hacker can see N, E and the encrypted number, it is extremely difficult to figure out the original credit card number without knowing the prime factors p and q.

Even trying every possible number on the clock face wouldn't work because N is so large, with over 100 digits, far more than atoms in the universe.

There is a "decoding number" D that allows the company to recover the original credit card number by raising the encrypted one to the power of D. But only the company knows D, since it depends on the secret primes p and q.

RSA issued a 129-digit number challenge to crack it. Most estimated it would take quadrillions of years but it was eventually cracked in 17 years, showing the system was not perfectly secure.

This drew more mathematicians to develop new techniques for factoring to crack RSA numbers, advancing the field of cryptography significantly.

Here is a summary:

Carl Pomerance is a mathematician who specializes in number theory and integer factorization. He became fascinated with factorization as a high school student when he failed to factor 8,051 in a competition.

This failure fueled his lifelong quest to develop fast factorization methods. He later learned the method his teacher had in mind, based on Fermat's factorization method.

When RSA published their 129-digit challenge number in 1977, Pomerance saw it as an opportunity to solve the problem that had vexed him as a student. He developed the quadratic sieve algorithm to efficiently factor large numbers.

In the early 1990s, Arjen Lenstra and Mark Manasse realized the Internet could be harnessed to distribute the workload of the quadratic sieve across many computers. They launched a project that cracked the RSA 129 number in 1994 using over 500,000 processors.

Newer algorithms like the number field sieve have since surpassed the quadratic sieve. The current record is factoring RSA 155. Businesses rely on these cryptosystems but many are unprepared for a potential mathematical breakthrough.

This passage discusses several topics related to internet security and prime numbers:

Internet business security is more dependent on human errors than mathematical flaws. Sites leave sensitive data unencrypted, and weak passwords/keys can be cracked.

During WWII, German Enigma codes were cracked due to operator errors, not mathematical flaws. Similarly, RSA could be weakened by poor random number generation for keys.

Finding large prime numbers is crucial for RSA encryption but becomes increasingly difficult. The Riemann Hypothesis, if proven true, could yield a fast algorithm for discovering primes.

There are likely enough large prime numbers for everyone to have strong encryption keys. Prime number distribution is well-understood thanks to theorems like the Prime Number Theorem.

Testing if a number is prime can be done efficiently using techniques like Fermat's Little Theorem and the Miller-Rabin primality test. These allow verifying primes without needing to fully factorize numbers.

While cracking encryption remains theoretically possible, the sheer scale of the problem and lack of proven algorithms give businesses confidence to rely on existing crypto systems for most security needs. Advancements in prime number theory could change that over time.

Here is a summary:

The Riemann Hypothesis is important for proving whether the Miller-Rabin primality test is efficient. Three Indian mathematicians devised an alternative test that did not assume the Riemann Hypothesis. This surprised the community and demonstrated Indians' continued contributions to number theory.

Cryptography relies on the difficulty of factoring large numbers into primes. So far, nature has not revealed any fast way to do this, helping cryptography. But this may not last forever.

The discovery of RSA cryptography raised the profile and funding of number theory research. Fields medalist Timothy Gowers used RSA as an example of mathematics' real-world impacts.

Koblitz and Miller independently proposed elliptic curve cryptography in the 1980s as a smaller and faster alternative to RSA suitable for mobile devices. It is now widely used for mobile commerce.

Koblitz remains passionate about pure mathematics but his work on elliptic curve cryptography led to business applications and government restriction due to national security interests.

Here is a summary:

Elliptic curve cryptography was initially seen as a threat to RSA's dominance in internet cryptography. RSA expressed doubts about elliptic curve security on its ECC Central website.

Neal Koblitz supported elliptic curve cryptography. When a critic compared understanding elliptic curves to understanding obscure Chaldean poetry, Koblitz had t-shirts made saying "I love Chaldean poetry" as a joke.

Elliptic curve cryptography has stood the test of time and been adopted in many standards and applications. However, in 1998 Joseph Silverman proposed a potential cryptanalysis technique based on heuristics from the unsolved Birch–Swinnerton-Dyer conjecture regarding elliptic curves.

The Birch–Swinnerton-Dyer conjecture relates properties of elliptic curves to hypothetical "landscapes" and could potentially provide clues to solve problems underlying elliptic curve cryptography if the conjecture is proven. Silverman's proposal raised concerns about elliptic curve security, though the attack was considered inefficient.

So in summary, it discusses the initial skepticism of elliptic curves from RSA, their adoption over time, but also how an unsolved mathematical conjecture briefly raised fears it could undermine elliptic curve cryptography security.

Here is a summary:

Hugh Montgomery was investigating an unrelated problem from his graduate studies about how imaginary numbers factorize, inspired by Gauss's work.

While studying this, he unexpectedly discovered patterns in the zeros of the Riemann zeta function along the critical line. This suggested the positions of the zeros along the line may be more ordered than previously thought.

His findings provided some of the best evidence to date supporting the truth of the Riemann Hypothesis, which states that all non-trivial zeros lie on the critical line.

Montgomery was originally just trying to extend the work of mathematician Alan Baker, who had made progress on one of Gauss's factorization problems. This led Montgomery down an unexpected path to discoveries related to the Riemann Hypothesis.

His orderly patterns for the zeros provided surprising evidence of order rather than randomness, going against common assumptions at the time about the distribution of zeros along the critical line. This supported the idea that a proof may be possible.

So in summary, Montgomery made unexpected but important contributions to understanding the Riemann Hypothesis while investigating an unrelated problem, discovering more order and pattern in the zeros than previously thought.

Here is a summary:

Montgomery returned to Cambridge reassured that understanding number theory was not futile. He focused on Gauss's unsolved problem of factorizing imaginary numbers.

The Riemann Hypothesis played a role in proving one of Gauss's conjectures, known as the Class Number Conjecture. In 1916, Hecke proved that if the RH is true, then Gauss's conjecture is also true. Later, others showed that if the RH is false, Gauss's conjecture can still be proven.

Montgomery hoped to use clusters of Riemann zeros to solve problems about factorizing imaginary numbers. However, his work using Hardy/Littlewood's twin prime conjecture surprisingly indicated that zeros repel each other, not cluster.

Montgomery created a graph showing the expected separation of zeros. It was unlike other distributions and suggested zeros are uniformly spaced. This did not help with his original goals.

Montgomery showed his work to Selberg, worried Selberg may have already known it. At a conference, Freeman Dyson helped Montgomery interpret the "message" in his unexpected discovery about the distribution of zeros.

Here is a summary:

Freeman Dyson became fascinated with mathematics as a young student, particularly number theory and Ramanujan's work. He saw himself as more of a "frog" who dove deep into specific areas, rather than a "bird" who saw broad connections.

Dyson went on to have a successful career in physics, promoting Feynman's work in quantum physics. He took a position at the Institute for Advanced Study in 1953.

In the 1970s, Montgomery met Dyson through number theorist Sarvadaman Chowla at Princeton. Montgomery was explaining his ideas about patterns in the distribution of zeros of the Riemann zeta function.

Dyson recognized these patterns as being similar to those found in the energy levels of nuclei in quantum physics. He explained the connection between Montgomery's work and random matrix theory used in physics.

When Montgomery looked at experimental data on energy levels in erbium, he saw a striking similarity to the patterns in Riemann zeros, suggesting the mathematics behind both phenomena could be linked. This meeting with Dyson helped Montgomery realize the potential connection between the Riemann hypothesis and quantum physics.

Here is a summary:

In the quantum world, observing a particle like an electron inevitably changes its behavior, unlike in the classical macro world where we can observe without interacting.

Before observation, quantum objects exist in an "imaginary" domain described by imaginary numbers, not the real numbers of classical physics. Observation causes this quantum world to "collapse" into the real domain.

Werner Heisenberg and Max Born helped develop the modern quantum model of the atom by studying energy levels mathematically through imaginary numbers and waves.

Determining energy levels becomes vastly more complex for heavier atoms with many particles. Eugene Wigner and Lev Landau took a statistical approach, finding energy level statistics often don't depend drastically on the precise shape or equations used.

Montgomery found Riemann zeta function zeros seem to behave similarly to energy level statistics. If proven, this could explain the Riemann Hypothesis by relating zeros to physical systems forbidden to have "imaginary" energies off the critical line.

Andrew Odlyzko used powerful computers to calculate extremely distant zeros, finding behavior matching Montgomery's predictions, providing experimental validation where mathematics alone could not yet reach.

Here is a summary:

Andrew Odlyzko continued his analysis of the Riemann zeta function, calculating the zeros up to 10^20. He found an even better statistical match between the spacings of the zeros and Montgomery's prediction, providing strong evidence the zeros are related to a quantum mechanical system.

Persi Diaconis, a master statistician and debunker of things like psychic phenomena, thoroughly tested the data and couldn't find a better statistical fit. However, more work was needed to conclusively determine the underlying mechanism.

Diaconis became interested in probability and statistics after starting as a magician's assistant at age 14. He developed techniques for detecting patterns and randomness.

Diaconis applied these techniques to problems like card shuffling and analyzing zeros of the zeta function. He confirmed the zeros matched predictions from random matrix theory.

Connections were found between the statistics of zeros, heavy nuclei, glass transitions, and other systems. These pointed to an underlying relationship with quantum physics.

Diaconis is still working on problems like determining average success rates in the card game Klondike, where the zeros may provide insight due to connections with random matrix theory. More work is needed to fully understand these relationships.

Here is a summary:

Odlyzko was doing statistical analysis of the Riemann zeta zeros and comparing them to predictions from quantum mechanics.

He noticed some discrepancies creeping into the data when analyzing a statistical measure called the number variance. The graphs for the Riemann zeros were starting to deviate from those predicted by random quantum systems like drums.

This showed that the early connection between Riemann zeros and quantum physics was breaking down. Odlyzko thought it may be due to computational errors.

Michael Berry realized these deviations matched what would be expected from chaotic quantum systems like chaotic billiards, where the path of particles is highly sensitive to initial conditions.

In chaotic quantum billiards, the statistics of energy level spacings further apart (like the Nth and N+1000th) differ from predictions of random systems. This matched what Odlyzko observed.

Berry's insights from chaos theory and quantum physics provided an explanation for the discrepancies - the Riemann zeros may behave like energies from a chaotic quantum system rather than a random one.

Here is a summary:

Berry recognized a connection between the statistics of quantum chaotic billiards and the zeros of the Riemann zeta function. This provided evidence that the Riemann zeros may have an underlying quantum mechanical origin related to quantum chaos.

If true, this could unite quantum physics, chaos theory, and prime numbers. The primes may correspond to periodic paths/orbits in a chaotic quantum billiard system.

Many physicists are now interested in searching for a physical system that exhibits this behavior and could explain the Riemann zeros. However, the mathematical explanation may not necessarily have a physical model.

Companies like AT&T and Hewlett-Packard supported this work due to its potential to improve understanding of quantum systems relevant to computing.

Fry Electronics also supports mathematical research through the American Institute of Mathematics next to its stores. This was launched by Brian Conrey and John Fry to coordinate efforts to solve the Riemann Hypothesis through collaboration.

At a meeting hosted by Fry Electronics, Peter Sarnak voiced skepticism about claims of a connection without clear mathematical evidence. He challenged physicists to use the quantum chaos analogy to provide new insights about the Riemann zeta function. Jon Keating would later win a prize for providing such insights related to the number 42.

Here is a summary of the key points:

Number theorist Conrey was working on calculating the moments of the Riemann zeta function, which should give a sequence of numbers, but they had only calculated the first few terms.

At a meeting with physicists, Conrey challenged them to explain why the next number in the sequence appeared to be 42 based on his work with Ghosh.

Physicist Jon Keating took up the challenge and worked with Nina Snaith to develop a formula that generated the whole sequence. It correctly produced 42.

Conrey and colleague Gonek independently guessed the next term would be 24,024. Just before Keating's lecture, the physicists and mathematicians confirmed his formula also produced 24,024, providing strong evidence it was correct.

Keating's work opened up a fruitful analogy between the Riemann zeta function and quantum chaos that helped mathematicians make progress.

It was later discovered that Riemann himself had used similar methods involving vibrations and frequencies to solve a classical hydrodynamics problem, unexpectedly connecting his early work to the physicists' approaches.

So in summary, the physicists' analogy helped explain mysterious constants in the Riemann zeta function and sparked new insights, highlighting connections between Riemann's own work and modern approaches.

Here is a summary:

André Weil, a prominent young French mathematician, sent a paper from a military prison in Rouen in 1940 to the prestigious French mathematics journal Comptes Rendus.

To the editor Elie Cartan's surprise, the paper proposed a new technique to show why points at sea level in certain mathematical landscapes tend to lie on a straight line. While it didn't work for Riemann's zeta function landscape, it was significant progress.

Weil's ideas were developed further by mathematician Alain Connes and influenced his approach to the Riemann Hypothesis.

Weil ended up in prison due to his obsession with ancient languages like Sanskrit. He believed sophisticated mathematics developed alongside language. His linguistic skills helped him develop new mathematical languages.

Weil's incarceration prompted him to make an important breakthrough while in solitary confinement, building on his earlier work and influencing later mathematicians working on landscapes like Riemann's. His work was part of a resurgence of French mathematics after WWII.

Here is a summary:

André Weil was inspired to pursue mathematics after studying Gauss's Prime Number Theorem. He entered the prestigious École Normale Supérieure in Paris at age 16 to begin his formal mathematical training.

At the École, Weil also indulged his passion for ancient languages like Sanskrit. He became fascinated with Hindu epics like the Bhagavad Gita. This inspired his belief that one should read original source texts to fully appreciate ideas.

After completing his exams, Weil traveled across Europe attending lectures by leading mathematicians. At Göttingen, he began developing ideas for his doctoral thesis.

Weil was part of a group of young French mathematicians called Nicolas Bourbaki who wanted to systematically organize and present contemporary mathematics to revive French prominence in the field.

Weil's interest in Eastern philosophy influenced his pacifist views. During World War II, he fled France to avoid conscription but was imprisoned in Finland on suspicion of spying before being released.

Imprisoned again in France for desertion, Weil continued his mathematical work, making advances related to zeta functions that helped explain patterns in number theory, even if not solving the Riemann Hypothesis. His work laid important foundations.

Here is a summary:

André Weil proved a theorem while imprisoned in Rouen during WWII that showed points of certain mathematical landscapes related to equations lie on a straight line, approaching a goal set by Riemann.

This was an important breakthrough, though not a solution to Riemann's hypothesis directly. It established techniques in the new field of algebraic geometry.

Weil framed the solutions to equations as frequencies of a mathematical "drum" he constructed. He realized he could force the frequencies to align using prior work by Guido Castelnuovo.

After release, Weil was unable to match this success in extending the results to Riemann's landscape for prime numbers. He expressed regret at never solving or seeing a proof of the Riemann hypothesis.

However, his work established algebraic geometry and gave mathematicians hope and evidence that Riemann was correct. It provided a framework that future mathematicians like Grothendieck would build on in attempts to prove the Riemann hypothesis.

So in summary, Weil made an important breakthrough while imprisoned but was unable to fully solve the Riemann hypothesis, though his work inspired future efforts and helped establish algebraic geometry.

Here is a summary:

In the mid-20th century, there was an effort in France to re-establish Paris as a center of mathematical activity, after it had lost prominence to Göttingen.

The Institut des Hautes Études Scientifiques (IHÉS) was founded in 1958 outside Paris, modeled after the Institute for Advanced Study in Princeton. It aimed to be independently funded and not controlled by the state.

Alexandre Grothendieck was an early and influential professor appointed at IHÉS. He developed a revolutionary new language for mathematics that allowed expressing previously inexpressible ideas. This transformed fields like algebraic geometry.

IHÉS became central to the Bourbaki group's effort to publish comprehensive treatises on modern mathematics. Grothendieck was a major contributor.

Some criticized Bourbaki for presenting mathematics as a finished product rather than evolving organism. Grothendieck saw it as confirming the modern foundation, like Euclid's Elements.

Grothendieck made huge advances but his radical new language was difficult for others to grasp. He failed to resolve the Riemann Hypothesis.

Grothendieck's strong pacifist political views eventually led him to resign from IHÉS in 1970 when he learned of its military funding connections. He then largely withdrew from mathematics.

Here is a summary:

Alexander Grothendieck was obsessed with completing his mathematical vision but was unable to do so, which led to bitterness. He violently attacked those who developed his work in new directions.

Grothendieck now lives remotely in the Pyrenees and believes the devil is destroying divine harmony in the world, showing signs of mental decline. Attempting to prove the Riemann Hypothesis also drove John Forbes Nash mad.

Alain Connes developed the field of non-commutative geometry, going beyond previous work. He believed he identified a complex geometric space using strange p-adic numbers that could naturally produce the Riemann zeros and solve the hypothesis.

Connes is an explorer who strikes out into new mathematical territories beyond the current horizon. While others stay close to familiar landscapes, he voyages across uncharted waters. His work was hoped to integrate various mathematical strands and finally solve the Riemann Hypothesis.

Here is a summary:

Alain Connes believed he had made significant progress toward solving the Riemann Hypothesis through his approach of using non-commutative geometry. However, he knew there was still work to be done to fully verify his ideas.

In 1997, Connes went to Princeton to present his ideas to some of the top experts in the field: Bombieri, Selberg, Sarnak, and Katz. Princeton was still considered the leading center for work on the Riemann Hypothesis. These mathematicians would provide a rigorous examination of Connes' ideas.

They felt Connes had made some progress in developing earlier ideas, but still had problems to address. Specifically, his approach seemed to make any points off Riemann's conjectured line "disappear". More work was needed.

Bombieri eventually played an April Fools prank, announcing Connes had solved it, generating excitement before people realized it was a joke. This event marked the end of the initial excitement about Connes' approach.

Years later, while progress has been made, Connes' ideas have not yet yielded a full solution. The Riemann Hypothesis remains unproven, demonstrating the immense difficulty of the problems surrounding prime numbers and their patterns. Connes remains hopeful that further insights may one day solve this century-old challenge.

Here is a summary:

The story is about the development of the prime numbers in mathematics and their connections to other fields like technology, computer science, and physics.

Technological advances like computers have allowed mathematicians to study even larger prime numbers and see patterns that were previously impossible to observe.

Computer languages and the rise of Internet technology have brought prime numbers into mainstream relevance due to their role in encryption and computer security.

Mathematicians have made connections between prime numbers and other areas like quantum physics. Understanding primes has also led to new ideas in pure mathematics.

One major open problem is the Riemann Hypothesis, concerning the distribution of prime numbers. Many top mathematicians have tried and failed to prove it over the past century. Doing so could yield new insights.

The story traces how primes have spread from an abstract mathematical concept to having real-world importance today in technology and security due to these various breakthroughs, connections, and technological advances over time.

Here is a summary of the provided text:

The author spends several hours trying to understand and conceptualize the fourth dimension. They had support from the Royal Society, who provided funding as a Royal Society Research Fellow. This allowed the author to pursue mathematical ideas and communicate their work to broader audiences. Key teachers and schools also supported the author's education and inspiration to become a mathematician. Friends and family additionally provided encouragement and assistance during the writing process. The author thanks various individuals and organizations who helped make the book possible through editorial support, publishing opportunities, and general encouragement at different stages.

Here are summaries of the selected articles and books:

'Mordell’s review, Siegel’s letter to Mordell, Diophantine geometry, and 20th century mathematics’ (Notices of the American Mathematical Society, 1995) - The article discusses L.J. Mordell's review of C.L. Siegel's book on Diophantine geometry and Siegel's response letter. It places their discussion in the context of 20th century developments in mathematics.

Bernhard Riemann, 1826–1866: Turning Points in the Conception of Mathematics (1999) - A translated biography of Bernhard Riemann focusing on turning points in his work that changed mathematical conceptions.

'Noncommutative geometry' (Notices of the American Mathematical Society, 1997) - An article introducing the concepts and developments in noncommutative geometry.

A Mathematician’s Miscellany (1953) - A collection of essays by mathematician J.E. Littlewood on various mathematical topics.

'The Riemann hypothesis' (1962) - A short essay by J.E. Littlewood discussing the Riemann hypothesis.

'Mathematics at Göttingen under the Nazis' (Notices of the American Mathematical Society, 1995) - A piece on the impact of Nazi rule on mathematics teaching and research at the University of Göttingen.

'A brief report on a number of recently discovered sets of notes on Riemann’s lectures' (Historia Mathematica, 1988) - A report on newly discovered notes from Riemann's lectures and the transmission of his unpublished works.

Hilbert (1970) - A biography of mathematician David Hilbert.

Julia, A Life in Mathematics (1996) - A biography of mathematician Julia Robinson containing contributions from several authors.

The Elements of the Theory of Algebraic Numbers (1910) - A textbook on algebraic number theory introduced by David Hilbert.

The New Book of Prime Number Records (1996) - A book collecting prime number records.

The summaries briefly describe the main topics, focuses or contributions of each selected source based on their titles or other contextual information provided. Let me know if you need any part summarized in more detail.

Here is a summary of the key points:

The Cold War and developments in computing drove advances in cryptography during the 20th century. Important figures included Cole, Diffie, Hellman, and the NSA.

Gauss made breakthroughs in number theory, discovering patterns in primes and developing novel methods. He was influential but failed to widely disseminate his work. Dirichlet later continued his number theory work.

Hilbert addressed the International Congress of Mathematicians in 1900 and posed 23 influential problems. He brought mathematicians to Göttingen and worked on geometry, equations, and the Entscheidungsproblem.

Hardy and Littlewood collaborated on analytic number theory. Hardy worked with Ramanujan and believed the purity of mathematics was its own reward.

Gödel proved his incompleteness theorems showing the limits of axiomatic systems like those used in Hilbert's program.

Computers advanced dramatically from the earliest models to Cray machines. Figures included Babbage, Turing, von Neumann, and Lehmer. Programming languages also developed.

Cryptographic advances included elliptic curves, RSA, digital signatures, and quantum computing approaches. Figures included Rivest, Shamir, Adleman, Koblitz, Lenstra, and Shor.

Fields Medals highlighted early career achievement in areas like number theory, algebra, and topology. Prominent recipients included Fermat, Gauss, Hilbert, Gödel, Grothendieck, Wiles, and others.

Here is a summary of the key points about Marcus du Sautoy from the passage:

Marcus du Sautoy is a prominent British mathematician and Professor at Oxford University. He is known for his work on prime numbers and book "The Music of the Primes".

Unlike some mathematicians, he is passionate about communicating mathematics to non-specialists through public lectures, articles, radio appearances, and school visits.

He disagrees with G.H. Hardy's view that writing about mathematics is a sign of weakness for mathematicians past a certain age. Du Sautoy believes communication is an important part of his job.

He enjoys his work pursing pure mathematics from home, but also enjoys sharing his passion for numbers with a wider audience in simpler terms.

He sees himself as very fortunate to be paid to follow his passions for mathematics.

He credits his inspiration and passion for numbers to an influential schoolteacher who introduced him to mathematical puzzles and problems at a young age.

In summary, du Sautoy is a prominent mathematician who goes against the more insular views of some in his field by vigorously communicating and promoting mathematics to non-specialists.

Here is a summary:

Marcus du Sautoy wonders how many people might have been interested in math but were not given the opportunity due to bad teaching or lack of encouragement. Many adults may have looked away from math at a crucial time and never got a second chance.

Du Sautoy offers a metaphor comparing math to music. While composers understand complex structures, listeners don't need that expertise - they can simply listen and enjoy. Similarly, du Sautoy argues we can develop "mathematical ears" and appreciate the beauty of math without understanding every step.

He uses the example of Jerzy Grotowski, a 20th century theatre director known for his avant-garde, experimental work. Grotowski's theatre, like math, was intellectual, elite and innovative but appealed to very small audiences. Both demand great discipline from practitioners.

However, unlike math which aims to clearly explain a proof, theatre deliberately aims to be ambiguous and leave different impressions with each audience member. Only bad art or plays lack ambiguity. Good math is unambiguous in successfully explaining an idea, while good art allows for various interpretations.

Here is a summary of how they spend their days:

Marcus du Sautoy is one of several scientists mentioned. As a mathematician, he likely spends his days researching mathematics, working on proofs and theorems. He writes books and articles about mathematics to educate others. As a professor at Oxford University, he also spends time teaching students and working with colleagues.

The others mentioned include a cosmologist, meteorologist, and marine biologist. A cosmologist studies the universe - they likely spend their days analyzing astronomical and physics data, developing theories about the origins and structure of the universe. A meteorologist studies weather and climate - they likely forecast weather patterns, analyze weather data and trends, and research how weather systems work. A marine biologist studies ocean life - they likely conduct field research, examine marine specimens in labs, and work to understand ocean ecosystems and environmental issues.

In summary, they all spend their days immersed in scientific research through analyzing data, developing theories, conducting experiments or field work, teaching, publishing work, and collaborating with other scientists in their fields. Mathematics, astronomy, weather, and ocean life are all subjects that require dedicated study and investigation.