Prime numbers, crystals share similar structural patterns

According to a new study, the distribution of prime numbers is similar to the positioning of atoms inside some crystalline materials.

When scientists at Princeton University compared the pattern of prime numbers along a lengthy line of numbers with the atomic patterns revealed when crystals are blasted with X-rays, they were surprised by the similarities.

"There is much more order in prime numbers than ever previously discovered," Salvatore Torquato, professor of chemistry and the Princeton Institute for the Science and Technology of Materials, said in a news release. "We showed that the primes behave almost like a crystal or, more precisely, similar to a crystal-like material called a 'quasicrystal.'"

Until recently, mathematicians thought prime numbers, numbers divisible only by themselves and one, were scattered sporadically throughout the number line. But new research suggests there are patterns to be found when primes are analyzed at greater scales.

Research suggests prime number patterns resemble "hyperuniformity" patterns found in crystals, quasicrystals and other disordered systems. Hyperuniformity describes patterns that reveal themselves at large scales.

One way to identify hyperuniformity in crystals is to blast them with X-rays, a process known as crystallography. When an X-ray travels through a crystal's 3D atomic lattice, the light produces a pattern of bright spots called Bragg peaks.

When X-rays pass through quasicrystals, the resulting pattern of Brag peaks is more complex. In between the main Bragg peaks are additional Bragg peaks -- patterns within patterns.

When scientists designed a model to convert the pattern of prime numbers into a crystalline atomic structure -- into particles -- they found the theoretical quasicrystal produced Bragg-like peaks similar to the hyperuniformity patterns revealed by real quasicrystals.

The comparison only works if a sufficiently large portion of the number line is translated. Over shorter intervals, the pattern of prime numbers appears random and disordered.

"When you go to that distinguished limit, 'Boom!'" Torquato said. "The ordered structure pops out."

Researchers hope the findings -- detailed this week in the Journal of Statistical Mechanics -- will offer new insights into both mathematics and material science.

"Prime numbers have beautiful structural properties," said Torquato. "The primes teach us about a completely new state of matter."

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