An AI-produced proof inspired a mathematician to find a simpler argument about one of mathematics’ deepest mysteries. The Riemann hypothesis predicts that every non-trivial zero of the zeta function lies on the line Re(s) = ½. In a September 2026 preprint, Youness Lamzouri gives a conceptually simpler proof that more than 67.25% of these zeros are both on that line and simple, meaning they are not repeated roots. This is an asymptotic proportion, not a count from a finite computer search. An internal research version of Claude had produced an earlier proof of this bound, subsequently checked by mathematicians Levent Alpöge and Ralph Furman. Lamzouri replaced its intricate matrix framework with a single Hilbert space inequality. His approach also yielded additional estimates. This does not prove the Riemann hypothesis. Even a proportion of 100% in the asymptotic sense could leave exceptions. What makes this story significant is the passage from obtaining a result to understanding why it works. A clearer proof can become the starting point for further mathematics. Source: Youness Lamzouri, A new proof that more than 2/3 of the zeros of the Riemann zeta function are simple and on the critical line, arXiv:2609.02882.

Write 1, 2, 3, … in a square spiral. Mark only the primes. Unexpected diagonal streaks begin to appear. In 1963, Stanisław Ulam noticed this while doodling during a scientific talk. The resulting picture became known as the Ulam spiral. The primes look irregular along the number line, yet this simple rearrangement reveals striking visual structure.

Henry Briggs was so impressed by John Napier’s invention of logarithms that he travelled from London to Edinburgh simply to meet him. When they finally came face to face, the two mathematicians reportedly stood looking at one another for nearly fifteen minutes without speaking. Briggs had come to understand how Napier had conceived an idea that now seemed astonishingly natural. He then stayed at Napier’s home for a month. Mathematical Circles Sometimes a mathematical idea is so beautiful that the first response is silence. Source: Howard W. Eves, In Mathematical Circles.

One of mathematics’ strangest objects is the Monster group. It has more than 10⁵³ elements, yet it can be realized through symmetries in a space of 196,883 dimensions. Its exact order is a 54-digit integer. It's power of abstract algebra that produces objects whose sheer scale seems almost beyond imagination.

G. H. Hardy once explained mathematical intuition with a kite. Watching a boy fly one, Hardy told Ulam that you cannot see the wind, but you can feel the pull on the string. In much the same way, a mathematician may sense the direction of a problem before having a complete proof. A beautiful description of intuition: the invisible force comes first, the formal mathematics follows. Source: S. M. Ulam, Adventures of a Mathematician

A wheel does not have to be circular to roll smoothly. The Reuleaux triangle, built from three circular arcs, has constant width: the distance between two parallel supporting lines is the same in every direction. So although it looks like a rounded triangle, it can roll between parallel surfaces without making the upper surface rise and fall. This same geometry even helps create drill mechanisms capable of producing nearly square holes. Source: Erik Seligman, Math Mutation Classics.

Two words used constantly in modern mathematics carry the name of a 9th-century scholar. Around 825, al-Khwarizmi wrote Kitab al-jabr wa al-muqabala. From al-jabr came our word algebra. And from a Latinized form of al-Khwarizmi came algorithm.

Every triangle hides an equilateral triangle, even when the original triangle is completely irregular. Take any triangle. Divide each of its three angles into three equal parts. Now consider the intersections of the adjacent internal angle trisectors. The three intersection points always form an equilateral triangle. No condition is required: The original triangle may be acute, obtuse, scalene or isosceles. Its sides may have any lengths. Its angles may be completely different. Yet the hidden triangle always has three equal sides and three 60° angles. This is Morley’s trisector theorem. Local divisions of three unrelated angles produce perfect global symmetry.

Baseball once gave number theory a new pair of numbers. Babe Ruth’s career record stood at 714 home runs. On 8 April 1974, Hank Aaron hit number 715 and surpassed it. Now factor the two consecutive numbers: 714 = 2 × 3 × 7 × 17 715 = 5 × 11 × 13 Add their prime factors: 2 + 3 + 7 + 17 = 29 5 + 11 + 13 = 29 So 714 and 715 have exactly the same sum of prime factors. This coincidence inspired the term Ruth-Aaron pair for consecutive integers whose prime factors have equal sums. A sporting record changed from 714 to 715, and mathematics found a pattern connecting them.

Pierre de Fermat was not a professional mathematician. He was a lawyer and magistrate, had no particular formal mathematical training, and apparently did not become seriously interested in mathematics until after age 30. Yet mathematics was his “hobby.” He went on to help create analytic geometry, lay foundations for calculus and probability, and transform number theory. One of history’s most extraordinary mathematical amateurs. Source: David M. Burton, The History of Mathematics

At just 18, Gauss made a discovery that helped decide his future. For more than 2,000 years, no new constructible regular polygon with a prime number of sides had been found. Then, in March 1796, Gauss proved that a regular 17-gon could be constructed using only straightedge and compass. He began his famous mathematical diary soon afterward. Source: Boyer & Merzbach, A History of Mathematics

One of the strangest bridges in mathematics connects the Riemann zeta function to quantum physics. Andrew Odlyzko computed huge collections of non-trivial zeta zeros and studied the gaps between them. Their normalized spacings matched the statistical pattern of eigenvalues from GUE random matrices, objects arising in quantum theory. Number theory and quantum mechanics, apparently unrelated subjects, were displaying the same statistical fingerprint.

Nicholas Saunderson lost his sight to smallpox as a child, yet in 1711 he became the fourth Lucasian Professor of Mathematics at Cambridge. He lectured on optics, light, colours and vision, and developed a tactile “pin-board” for calculation and geometry: pegs placed in a grid represented digits, while threads stretched between them formed geometric figures. A remarkable example of mathematics being understood through structure, touch and abstraction rather than sight. Source: David Wells, The Penguin Book of Curious and Interesting Mathematics.

None of these points move in circles, so why does the flower rotate? Each dot only slides back and forth along its own straight line, following r = R cos(4θ − ωt). Because every line has a slightly different phase, the dots together form an 8-petal rose that turns at ω/4. No particle rotates, yet the pattern does. Math can make motion from phase.

During the Descartes–Fermat feud over optics, Fermat attacked Descartes’ derivation of refraction. He later approached the problem through his principle of least time and recovered the sine law of refraction by a different route. His method also implied something striking: light travels slower in water than in air. Later physics confirmed Fermat’s side of that dispute. Source: Hal Hellman, Great Feuds in Mathematics

A number can be mathematically happy. Take a positive integer, replace it by the sum of the squares of its digits, and repeat. For 28: 28 → 2² + 8² = 68 68 → 6² + 8² = 100 100 → 1 So 28 is a happy number. But unhappy numbers eventually fall into the cycle 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4.

From spirals to patterns, mathematics keeps finding its way into beauty.

Differentiation gave mathematics a language for changing slopes. Integration gave it a language for accumulating quantities and areas. Together, they transformed the study of motion, change, and modern analysis.

Some equations do more than solve a problem. They connect entire areas of mathematics and physics. Cédric Villani describes the Boltzmann equation as a meeting point of statistical physics, the arrow of time, fluid mechanics, probability, information theory, Fourier analysis, and more. A single equation can become a whole mathematical universe. Source: Cédric Villani, Birth of a Theorem.

Cut a circle into thin rings, straighten each one, and stack them. They form a triangle with base 2πr and height r. Its area is ½ · 2πr · r = πr². 🔵

On 16 January 1913, an unknown clerk in Madras sent G. H. Hardy a letter that would alter mathematical history. Ramanujan introduced himself as a man without university training who had spent his spare time developing mathematics independently. He boldly told Hardy that he was “striking out a new path for myself” and enclosed results on divergent series and prime numbers. A mathematician almost completely outside the academic world had decided to write directly to one of Cambridge’s leading figures. Source: Robert Kanigel, The Man Who Knew Infinity.

Consecutive Fibonacci numbers give a surprisingly good miles-to-kilometres approximation. For example: 8 miles ≈ 13 km 13 miles ≈ 21 km 34 miles ≈ 55 km Why? Because the ratio of consecutive Fibonacci numbers approaches the golden ratio, φ ≈ 1.618, which is remarkably close to 1 mile ≈ 1.609 km. A beautiful meeting of number theory and everyday estimation.

Kepler’s Second Law says that a line joining a planet to the Sun sweeps out equal areas in equal intervals of time. So a planet moves faster near perihelion and slower near aphelion. The speed changes, but the rate at which orbital area is swept out remains constant. A beautiful geometric signature of angular momentum conservation.

From history graduate to quantum pioneer Louis de Broglie did not begin his university life as a physicist. He first studied literature and history, earning a history degree in 1910. Only afterward did his growing fascination with science pull him toward physics. During World War I, he was assigned to the French Army’s wireless section and stationed at the Eiffel Tower, where he spent spare time studying technical and scientific problems. After the war he returned to theoretical physics, eventually developing the revolutionary idea that particles such as electrons possess wave-like properties. Source: Louis de Broglie – Biographical. Nobel Prize

A beautiful extension of amicable numbers is a sociable cycle. For example starting with 12496, repeatedly take the sum of proper divisors. After five steps, the sequence returns to where it began. This 5-cycle was discovered by Paul Poulet in 1918. Perfect numbers are cycles of length 1, amicable pairs are cycles of length 2, and sociable numbers continue the idea to longer loops.

Love behaves like a nonlinear system: small changes can create large effects, and the final outcome is not always easy to predict.

This formula produces a prime for every positive integer n. This is Mills’ theorem (1947). It sounds like a perfect prime-generating machine, but there is a catch: the constant must be known with increasing precision as the numbers explode in size. Therefore such formulas are considered as mathematical curiosities rather than practical ways of finding primes. Source: The Book of Prime Number Records (1996).

One of the most beautiful formulas for π is painfully slow. You need roughly 600 terms just to get π correct to two decimal places. James Gregory found this series in 1671, and Leibniz independently discovered it in 1674.

John Nash arrived at Carnegie Tech intending to become an engineer. Within two years, that plan had vanished. Mechanical drawing frustrated him. Chemistry bored him. Mathematics did neither. His professors quickly noticed the difference. One called him “a young Gauss,” and by his second year Nash had shifted almost entirely into mathematics, beginning the path that would define his life. Source: Sylvia Nasar, A Beautiful Mind.

Since π isn't a rational number, so never line up again, and the path never repeats.

Even a mathematician’s footnotes can become the main event. James Joseph Sylvester once prepared a 400-line poem, all rhyming with Rosalind, to illustrate his theory of versification. At the public reading, he announced that he would first explain the footnotes so the poem would not be interrupted. The footnotes led to digressions. An hour passed. Then an hour and a half. Only after noticing the time did Sylvester finally begin the poem. Source: David Wells, The Penguin Book of Curious and Interesting Mathematics.

If your Department of Mathematics had these faculty members, which papers would you choose?

Paul Halmos arrived in Chicago as a teenager speaking no English. He began college in chemical engineering, switched to mathematics, and later joked that he had been only a “routine calculus student.” Then came the remarkable turn: after earning his PhD, he typed 120 job applications and received just two replies, both saying no. He went to Princeton anyway, borrowing money from his father, and soon became John von Neumann’s assistant. Halmos later became one of mathematics’ great expositors. A career that began with rejection became a lesson in persistence, clarity, and curiosity. Source: MacTutor History of Mathematics.

In 1922, Walther Gerlach had a problem. His famous atomic-beam experiment had worked, but the evidence was almost invisible. A beam of silver atoms had passed through a magnetic field and struck a detector plate, leaving only the faintest traces of deposited silver. Gerlach struggled to see them. Then Otto Stern leaned over the plate. Stern was smoking one of his cheap cigars. The smoke contained sulphur compounds. When they reached the silver on the plate, they reacted with it, darkening the deposits into visible silver sulphide. Suddenly, the hidden pattern appeared. The experiment had revealed the striking splitting of the atomic beam that would become one of the iconic results of early quantum physics. And, remarkably, a cheap cigar helped make the evidence visible. Source: Walter Gratzer, Eurekas and Euphorias: The Oxford Book of Scientific Anecdotes.

As a teenager, Richard Feynman invented his own symbols for sine, cosine, tangent, logarithms, even derivatives. He thought they were as good as the standard notation. Then, while explaining mathematics to another student, he suddenly realized the problem: mathematics is not only about thinking clearly. It is also about communicating clearly. So he abandoned his private notation and returned to the standard symbols everyone could understand.

When you spend the whole day engaging with math accounts on X:

An eighteenth-century mathematician, working quietly on an abstract problem in probability, could never have imagined neural networks, recommendation systems, medical imaging, or the enormous computational models of today. His name was Thomas Bayes. An English minister with a deep interest in mathematics and probability, Bayes developed an idea whose influence would reach far beyond his own century. Yet the work that made his name famous was never published during his lifetime. Bayes died in 1761. After his death, his friend Richard Price discovered the manuscript, edited it, and presented it to the Royal Society. In 1763, it appeared under the modest title An Essay towards solving a Problem in the Doctrine of Chances. Hidden inside was an idea that would change the way we reason about uncertainty. The question was simple, but profound. Suppose we observe an event. What can that observation tell us about the unknown circumstances that produced it? Ordinary probability often begins with a process and asks what outcomes are likely. Bayesian reasoning turns the direction around. Start with what you believe. Observe the evidence. Then update that belief. Bayes explored this idea through an imagined experiment involving a table and balls. From that simple mathematical setting grew the foundations of what we now call Bayesian inference. In modern notation, the central relationship is beautifully compact: P(A|B) = P(B|A)P(A) / P(B) But the real beauty is not only in the formula. It is in the way of thinking behind it. Begin with what you know. Look at the new evidence. Revise what you believe. Again and again. Today, that principle appears across statistics, medical diagnosis, weather forecasting, signal processing, scientific inference, machine learning, and artificial intelligence. There is something remarkable about that journey. A problem written down in the eighteenth century became a fundamental tool for reasoning in the twenty-first. Thomas Bayes never lived to see his essay published. He could never have seen the technologies his mathematics would later help shape. Yet more than two and a half centuries later, we are still doing exactly what his mathematics teaches us to do: observe, update, and learn.

The geometry is identical; only the coordinate language changes.

A real professor makes a simple problem look complicated, just to impress the students.😄

When Maryam Mirzakhani received an email in 2014 informing her that she would receive the Fields Medal, her initial reaction was to assume that the sender’s account had been hacked. Source: Klarreich’s 2014 Quanta profile.

Immanuel Bloch and Jun Ye share the 2026 Wolf Prize in Physics for advances in controlling ultracold atomic systems. Bloch helped turn atoms trapped in lattices of laser light into experimental models of quantum matter. Ye developed optical atomic clocks of extraordinary precision. Here is a detail I find remarkable: in 2022, Ye’s team measured the difference in the rate of time between the top and bottom of an atomic cloud just a millimetre tall. The higher atoms ticked slightly faster, as Einstein’s general relativity predicts. As a math student, I find this deeply satisfying. An equation on a page becomes a measurable difference within a tiny cloud of atoms.

Functions transform, stretch, shift, and reflect. This is the geometry of mathematical dance.

At seventeen, John von Neumann was preparing to study chemical engineering in Berlin and Zurich. At the same time, he enrolled at Budapest University for an advanced degree in mathematics. His proposed PhD topic was nothing modest: the axiomatization of Cantor’s set theory, one of the most difficult and controversial areas of mathematics at the time. So while training for a practical profession, he was already attacking the foundations of modern mathematics. john-von-neumann Source: Norman Macrae's, John von Neumann.

A black hole becomes larger as its mass increases, because its Schwarzschild radius is directly proportional to mass. But its Hawking temperature behaves in exactly the opposite way. It decreases as the mass increases. So, rₛ ∝ M, while T ∝ 1/M. A more massive black hole is larger, yet colder.

We often call Euler’s identity the most beautiful equation in mathematics. But sometimes we forget the beauty of the Gaussian integral. It is absolutely gorgeous. And if you don’t know this integral yet, can you really call yourself a math nerd? 😉

Two bodies under Newtonian gravity can be beautifully simple: a planet and the Sun trace an ellipse. Add just one more body, and the mathematics changes dramatically. The three-body problem has no comparable tidy formula for the general motion. Its dynamics can become chaotic, so tiny differences in initial conditions may eventually lead to very different trajectories. This difficulty helped Poincaré develop a new way to study differential equations through the geometry and topology of their solutions, ideas that became foundational to modern nonlinear dynamics.

Every positive integer hides a compact formula for the sum of all its divisors.

When Stirling’s formula is taken a little too seriously.

There are infinitely many triples of consecutive triangular numbers whose product is a perfect square.