Prime numbers aren’t so random after all

They actually have a quasicrystal-like structure

  • Large prime numbers occur in a natural-looking pattern
  • The apparent randomness of prime numbers have long fascinated mathematicians
  • A thrilling discover that ties math and nature together

They've long fascinated mathematicians: prime numbers. They're numbers indivisible by any number other than themselves or 1, and they occur ever-more randomly as numbers increase in value. As mathematician R.C. Vaughan put it: "It is evident that the primes are randomly distributed but, unfortunately, we do not know what 'random' means."

Or at least they've always seemed to be random since the ancient Greeks first identified them. Now, theoretical chemist Salvatore Torquato of Princeton has discovered something startling: Large prime numbers actually occur according to a pattern that resembles the atomic structure of quasicrystals.

(Internet Archive Book Images)

1915 image of crystal structure revealed by x-rays

Cryptography’s not-so-secret sauce

For modern cryptography, prime numbers' randomness is handy. The ubiquitous RSA encryption algorithm multiplies two very large random numbers in the knowledge that deriving the two original values from their product is a beast of a computational problem. There's no direct connection between Torquato's finding and the soundness of cryptography that employs primes — at least not yet. But if they're not really random, well, maybe down the line it will become a problem. But that's not the really interesting part.

Looking at primes in a different way

It was a chemist's hunch. In chemistry, it's common to analyze the atomic structure of matter by firing x-rays at it and observing the ways in which the x-rays bounce off the material's atoms. Different materials produce different x-ray diffraction patterns. Torquato started wondering if there was a way to apply this analytic method to numbers, and what he might see.

Torquato, with grad student Ge Zhang, modeled long prime sequences as one-dimensional strings of particles, with primes represented by small spheres off which x-rays would bounce. It turned out that sequences containing about a million primes — such as the series starting with 10,000,000,019 — were sufficient to generate a meaningful analysis without incurring too much statistical noise. When virtual X-rays were shot at the particles, Torquato and Zhang saw something no one had seen before: Patterns not unlike the ones produced by already-weird quasicrystals, but also different. Still, Microsoft mathematician Henry Cohn tells Quanta, "What's beautiful about this is it gives us a crystallographer's view of what the primes look like."

Quanta's article on the discovery includes a visual explanation of the ways different materials scatter x-rays.

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

Numbers made physical

The implication is mind-bending. It's that prime numbers — non-corporeal digits, after all — can be envisioned as a natural physical system and, as Torquato tells Quanta, "a completely new category of structures." While it's long been understood that math can represent and describe a range of natural phenomena and systems, this is the first time primes seem to be themselves one of those systems.

The finding falls in line with research into "aperiodic order" — non-repeating patterns — prompted by the discovery of quasicrystals. As mathematical crystallographer Marjorie Senechal notes speaking to Quanta, "Techniques that were originally developed for understanding crystals … became vastly diversified with the discovery of quasicrystals. People began to realize they suddenly had to understand much, much more than just the simple straightforward periodic diffraction, and this has become a whole field, aperiodic order. Uniting this with number theory is just extremely exciting."

For Torquato, wherever this leads is secondary. The main payoff is simply being able to get a peek at what goes on behind the curtain with prime-numbers. "I actually think it's stunning," he tells Quanta. "It's a shock."

​There are two kinds of failure – but only one is honorable

Malcolm Gladwell teaches "Get over yourself and get to work" for Big Think Edge.

Big Think Edge
  • Learn to recognize failure and know the big difference between panicking and choking.
  • At Big Think Edge, Malcolm Gladwell teaches how to check your inner critic and get clear on what failure is.
  • Subscribe to Big Think Edge before we launch on March 30 to get 20% off monthly and annual memberships.
Keep reading Show less

Why the ocean you know and love won’t exist in 50 years

Can sensitive coral reefs survive another human generation?

Videos
  • Coral reefs may not be able to survive another human decade because of the environmental stress we have placed on them, says author David Wallace-Wells. He posits that without meaningful changes to policies, the trend of them dying out, even in light of recent advances, will continue.
  • The World Wildlife Fund says that 60 percent of all vertebrate mammals have died since just 1970. On top of this, recent studies suggest that insect populations may have fallen by as much as 75 percent over the last few decades.
  • If it were not for our oceans, the planet would probably be already several degrees warmer than it is today due to the emissions we've expelled into the atmosphere.
Keep reading Show less

Why modern men are losing their testosterone

Research has shown that men today have less testosterone than they used to. What's happening?

Flickr user Tom Simpson
Sex & Relationships
  • Several studies have confirmed that testosterone counts in men are lower than what they used to be just a few decades ago.
  • While most men still have perfectly healthy testosterone levels, its reduction puts men at risk for many negative health outcomes.
  • The cause of this drop in testosterone isn't entirely clear, but evidence suggests that it is a multifaceted result of modern, industrialized life.
Keep reading Show less

Health care: Information tech must catch up to medical marvels

Michael Dowling, Northwell Health's CEO, believes we're entering the age of smart medicine.

Photo: Tom Werner / Getty Images
Sponsored by Northwell Health
  • The United States health care system has much room for improvement, and big tech may be laying the foundation for those improvements.
  • Technological progress in medicine is coming from two fronts: medical technology and information technology.
  • As information technology develops, patients will become active participants in their health care, and value-based care may become a reality.
Keep reading Show less