Most of the relevant mathematics has been seen before in other guises. “The primes are actually suggesting a completely different state of particle positions that are like quasicrystals but are not like quasicrystals,” Torquato said.Īccording to numerous number theorists interviewed, there’s no reason to expect the Princeton team’s findings to trigger advances in number theory. In the primes’ case, though, distances between peaks are fractions of one another, unlike quasicrystals’ irrationally spaced Bragg peaks. These have dimmer peaks between them corresponding to farther-apart pairs of primes, and so on in an infinitely dense nesting of Bragg peaks.ĭense Bragg peaks have been seen before, in the diffraction patterns of quasicrystals, those strange materials discovered in the 1980s with symmetric but nonrepeating atomic arrangements. 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. It consists of a periodic sequence of bright peaks, which reflect the most common spacings of primes: All of them (except 2) are at odd-integer positions on the number line, multiples of two apart. The Princeton researchers have dubbed the fractal-like pattern “effective limit-periodicity.” 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. “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 Massachusetts Institute of Technology. Although mathematicians have uncovered many rules over the centuries about the primes’ spacings, “it’s very difficult to find any clear pattern, so we just think of them as ‘something like random.’”īut in three new papers - one by Torquato, Zhang and the computational chemist Fausto Martelli that was published in the Journal of Physics A in February, and two others co-authored with de Courcy-Ireland that have not yet been peer-reviewed - the researchers report that the primes, like crystals and unlike liquids, produce a diffraction pattern. “They are in many ways pretty hard to tell apart from a random sequence of numbers,” de Courcy-Ireland said. Primes, the indivisible building blocks of all natural numbers, skitter erratically up the number line like the bounces of a skipping rock, stirring up deep questions in their wake. It wasn’t clear what kind of pattern would emerge or if there would be one at all. (They found that this “Goldilocks interval” contains enough primes to produce a strong signal without their getting too sparse to reveal an interference pattern.) In computer experiments, they bounced light off long prime sequences, such as the million-or-so primes starting from 10,000,000,019. Hoping to highlight the elusive order in the distribution of the primes, he and his student Ge Zhang had modeled them as a one-dimensional sequence of particles - essentially, little spheres that can scatter light. Torquato told de Courcy-Ireland, a final-year graduate student at Princeton who had been recommended by another mathematician, that a year before, on a hunch, he had performed diffraction on sequences of prime numbers. The spacing of these bright spots, known as “Bragg peaks” after the father-and-son crystallographers who pioneered diffraction in the 1910s, reveals the organization of the scattering objects. But the symmetrically arranged atoms in a crystal reflect light waves in sync, producing periodic bright spots where reflected waves constructively interfere. When hit with X-rays, disorderly molecules in liquids or glass scatter them every which way, creating no discernible pattern. In his field, a standard way to deduce structure is to diffract X-rays off things. About a year ago, the theoretical chemist Salvatore Torquato met with the number theorist Matthew de 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.Ī professor of chemistry at Princeton University, Torquato normally studies patterns in the structure of physical systems, such as the arrangement of particles in crystals, colloids and even, in one of his better-known results, a pack of M&Ms.
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