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https://www.quantamagazine.org/a-chemist-shines-light-on-a-surprising-prime-number-pattern-20180514/ May 14, 2018

A Chemist Shines Light on a Surprising PrimeNumber PatternWhen a crystallographer treated prime numbers as a system of particles, the resulting diffractionpattern created a new view of existing conjectures in number theory.

By Natalie Wolchover

Olena Shmahalo/Quanta Magazine

About a year ago, the theoretical chemist Salvatore Torquato met with the number theorist Matthewde Courcy-Ireland to explain that he had done something highly unorthodox with prime numbers,those positive integers that are divisible only by 1 and themselves.

A professor of chemistry at Princeton University, Torquato normally studies patterns in the structureof physical systems, such as the arrangement of particles in crystals, colloids and even, in one of hisbetter-known results, a pack of M&Ms. In his field, a standard way to deduce structure is to diffract

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X-rays off things. When hit with X-rays, disorderly molecules in liquids or glass scatter them everywhich way, creating no discernible pattern. But the symmetrically arranged atoms in a crystal reflectlight waves in sync, producing periodic bright spots where reflected waves constructively interfere.The spacing of these bright spots, known as “Bragg peaks” after the father-and-soncrystallographers who pioneered diffraction in the 1910s, reveals the organization of the scatteringobjects.

Torquato told de Courcy-Ireland, a final-year graduate student at Princeton who had beenrecommended by another mathematician, that a year before, on a hunch, he had performeddiffraction on sequences of prime numbers. Hoping to highlight the elusive order in the distributionof the primes, he and his student Ge Zhang had modeled them as a one-dimensional sequence ofparticles — essentially, little spheres that can scatter light. In computer experiments, they bouncedlight off long prime sequences, such as the million-or-so primes starting from 10,000,000,019. (Theyfound that this “Goldilocks interval” contains enough primes to produce a strong signal without theirgetting too sparse to reveal an interference pattern.)

It wasn’t clear what kind of pattern would emerge or if there would be one at all. Primes, theindivisible building blocks of all natural numbers, skitter erratically up the number line like thebounces of a skipping rock, stirring up deep questions in their wake. “They are in many ways prettyhard to tell apart from a random sequence of numbers,” de Courcy-Ireland said. Althoughmathematicians have uncovered many rules over the centuries about the primes’ spacings, “it’s verydifficult to find any clear pattern, so we just think of them as ‘something like random.’”

But in three new papers — one by Torquato, Zhang and the computational chemist Fausto Martellithat was published in the Journal of Physics A in February, and two others co-authored with deCourcy-Ireland that have not yet been peer-reviewed — the researchers report that the primes, likecrystals and unlike liquids, produce a diffraction pattern.

“What’s beautiful about this is it gives us a crystallographer’s view of what the primes look like,”said Henry Cohn, a mathematician at Microsoft Research New England and the MassachusettsInstitute of Technology.

C. Todd Reichart (Torquato and Zhang) and courtesy of Matthew de Courcy-Ireland

Left to right: Salvatore Torquato, Ge Zhang and Matthew de Courcy-Ireland have written new papers looking atprime number sequences as particles producing a diffraction pattern.

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https://www.quantamagazine.org/a-chemist-shines-light-on-a-surprising-prime-number-pattern-20180514/ May 14, 2018

The resulting pattern of Bragg peaks is not quite like anything seen before, implying that the primes,as a physical system, “are a completely new category of structures,” Torquato said. The Princetonresearchers have dubbed the fractal-like pattern “effective limit-periodicity.”

It consists of a periodic sequence of bright peaks, which reflect the most common spacings ofprimes: All of them (except 2) are at odd-integer positions on the number line, multiples of twoapart. Those brightest bright peaks are interspersed at regular intervals with less bright peaks,reflecting primes that are separated by multiples of six on the number line. These have dimmerpeaks between them corresponding to farther-apart pairs of primes, and so on in an infinitely densenesting of Bragg peaks.

Dense Bragg peaks have been seen before, in the diffraction patterns of quasicrystals, those strangematerials discovered in the 1980s with symmetric but nonrepeating atomic arrangements. In theprimes’ case, though, distances between peaks are fractions of one another, unlike quasicrystals’irrationally spaced Bragg peaks. “The primes are actually suggesting a completely different state ofparticle positions that are like quasicrystals but are not like quasicrystals,” Torquato said.

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https://www.quantamagazine.org/a-chemist-shines-light-on-a-surprising-prime-number-pattern-20180514/ May 14, 2018

Quanta Magazine

https://www.quantamagazine.org/a-chemist-shines-light-on-a-surprising-prime-number-pattern-20180514/ May 14, 2018

Quanta Magazine

https://www.quantamagazine.org/a-chemist-shines-light-on-a-surprising-prime-number-pattern-20180514/ May 14, 2018

Quanta Magazine

https://www.quantamagazine.org/a-chemist-shines-light-on-a-surprising-prime-number-pattern-20180514/ May 14, 2018

Lucy Reading-Ikkanda/Quanta Magazine; Crystal diffraction pattern by Sven.hovmoeller; Quasicrystal diffractionpattern by Materialscientist)

According to numerous number theorists interviewed, there’s no reason to expect the Princetonteam’s findings to trigger advances in number theory. Most of the relevant mathematics has beenseen before in other guises. Indeed, when Torquato showed his plots and formulas to de Courcy-Ireland last spring (at the suggestion of Cohn), the young mathematician quickly saw that the primediffraction pattern “can be explained in terms of almost universally accepted conjectures in numbertheory.”

It was the first of many meetings between the two at the Institute for Advanced Study in Princeton,N.J., where Torquato was spending a sabbatical. The chemist told de Courcy-Ireland that he coulduse his formula to predict the frequency of “twin primes,” which are pairs of primes separated bytwo, like 17 and 19. The mathematician replied that Torquato could in fact predict all otherseparations as well. The formula for the Bragg peaks was mathematically equivalent to the Hardy-Littlewood k-tuple conjecture, a powerful statement made by the English mathematicians GodfreyHardy and John Littlewood in 1923 about which “constellations” of primes can exist. One ruleforbids three consecutive odd-numbered primes after {3, 5, 7}, since one in the set will always bedivisible by three, as in {7, 9, 11}. This rule illustrates why the second-brightest peaks in the primes’diffraction pattern come from pairs of primes separated by six, rather than four.

Hardy and Littlewood’s conjecture further specified how often all the allowed prime constellationswill occur along the number line. Even the simplest case of Hardy-Littlewood, the “twin primesconjecture,” although it has seen a burst of modern progress, remains unproved. Because primediffraction essentially reformulates it, experts say it’s highly unlikely to lead to a proof of Hardy-Littlewood, or for that matter the famous Riemann hypothesis, an 1859 formula linking the primes’distribution to the “critical zeros” of the Riemann zeta function.

The findings resonate, however, in a relatively young research area called “aperiodic order,”essentially the study of nonrepeating patterns, which lies at the intersection of crystallography,dynamical systems, harmonic analysis and discrete geometry, and grew after the discovery ofquasicrystals. “Techniques that were originally developed for understanding crystals … becamevastly diversified with the discovery of quasicrystals,” said Marjorie Senechal, a mathematicalcrystallographer at Smith College. “People began to realize they suddenly had to understand much,much more than just the simple straightforward periodic diffraction,” she said, “and this has becomea whole field, aperiodic order. Uniting this with number theory is just extremely exciting.”

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Adapted from Parcly Taxel

The primes’ pattern resembles a kind of aperiodic order known since at least the 1950s called limitperiodicity, “while adding a surprising twist,” Cohn said. In true limit-periodic systems, periodicspacings are nested in an infinite hierarchy, so that within any interval, the system contains parts ofpatterns that repeat only in a larger interval. An example is the tessellation of a strange,multipronged shape called the Taylor-Socolar tile, discovered by the Australian amateurmathematician Joan Taylor in the 1990s, and analyzed in detail with Joshua Socolar of DukeUniversity in 2010. According to Socolar, computer experiments indicate that limit-periodic phasesof matter should be able to form in nature, and calculations suggest such systems might haveunusual properties. No one guessed a connection to the primes. They are “effectively” limit periodic— a new kind of order — because the synchronicities in their spacings only hold statistically acrossthe whole system.

For his part, de Courcy-Ireland wants to better understand the “Goldilocks” scale at which effectivelimit-periodicity emerges in the primes. In 1976, Patrick Gallagher of Columbia University showedthat the primes’ spacings look random over short intervals; longer strips are needed for their pattern

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to emerge. In the new diffraction studies, de Courcy-Ireland and his chemist collaborators analyzeda quantity called an “order metric” that controls the presence of the limit-periodic pattern. “You canidentify how long the interval has to be before you start seeing this quantity grow,” he said. He isintrigued that this same interval length also shows up in a different prime number rule calledMaier’s theorem. But it’s too soon to tell whether this thread will lead anywhere.

The main advantage of the prime diffraction pattern, said Jonathan Keating of the University ofBristol, is that “it is evocative” and “makes a connection with different ways of thinking.” But theesteemed number theorist Andrew Granville of the University of Montreal called Torquato andcompany’s work “pretentious” and “just a regurgitation of known ideas.”

Torquato isn’t especially concerned about how his work will be perceived by number theorists. Hehas found a way to glimpse the pattern of the primes. “I actually think it’s stunning,” he said. “It’s ashock.”