The seemingly random digits known as prime numbers are not nearly as scattershot as previously thought. A new analysis by Princeton University researchers has uncovered patterns in primes that are similar to those found in the positions of atoms inside certain crystal-like materials.
The researchers found a surprising similarity between the sequence of primes over long stretches of the number line and the pattern that results from shining X-rays on a material to reveal the inner arrangement of its atoms. The analysis could lead to predicting primes with high accuracy, said the researchers. The study was published Sept. 5 in the Journal of Statistical Mechanics: Theory and Experiment.
“There is much more order in prime numbers than ever previously discovered,” said Salvatore Torquato, Princeton’s Lewis Bernard Professor of Natural Sciences, professor of chemistry and the Princeton Institute for the Science and Technology of Materials. “We showed that the primes behave almost like a crystal or, more precisely, similar to a crystal-like material called a ‘quasicrystal.'”
Primes are numbers that can only be divided by 1 and themselves. Very large primes are the building blocks of many cryptography systems. Primes appear to be sprinkled randomly along the number line, although mathematicians have discerned some order. The first few primes are 2, 3, 5, 7 and 11, becoming more sporadic higher in the number line.
Torquato and his colleagues have found that that, when considered over large swaths of the number line, prime numbers are more ordered than previously believed, falling within the class of patterns known as “hyperuniformity.”
Hyperuniform materials have special order at large distances and include crystals, quasicrystals and special disordered systems. Hyperuniformity is found in the arrangement of cone cells in bird eyes, in certain rare meteorites, and in the large-scale structure of the universe.
The team showed that the order they found in the prime numbers maps to the pattern that results when X-rays interact with certain forms of matter. As a chemist, Torquato is familiar with X-ray crystallography, shining X-rays through a crystal’s three-dimensional atomic lattice. With diamonds or other crystals, this will result in a predictable pattern of bright spots or peaks, known as Bragg peaks.
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