Mathematicians have never shrunk from complexity, but a wizardly math professor and postdoc fellow from Stanford have just unbottled another confusing prime-number daemon, and this one is laughing at our hubris. Prime numbers near to each other tend not to want to echo each other’s last digits, the mathematicians say: For example, a prime that ends in 1 is less likely to be followed by another ending in 1 than one might expect from a random sequence, “preferring” instead to be followed by a prime ending in 3 or 7. It’s like an anti-pattern, injecting insolent disorder where there should be probabilistic order. “Every single person we’ve told this ends up writing their own computer program to check it for themselves,” lead
summoner author Kannan Soundararajan told Nature. So what’s going on under the hood here?
The short answer is that we still don’t know, and it’s partially because this discovery made things yet more ambiguous. There are rules governing the distribution and behavior of prime numbers, but just beyond the conclusively proven lurks a black box full of fascinating uncertainty. What defines a prime number is unambiguous: it’s a whole number that can’t be evenly divided by any number other than 1 and itself. (That also means a prime number can’t be even, and can’t end in 5.)
The Prime Number Theorem describes the way prime numbers occur with decreasing frequency as numbers approach infinity, but Euclid proved that there are infinitely many primes, so there is no largest prime number; the largest prime number we currently know of is a Mersenne prime, but those are a whole ‘nother kettle of analytical fish. Beyond that, though, there seems to be little structure that can help to predict where the next prime will occur. We’ve had some fascinating insights on the problem, including one mathematician who lectures in simple, poetic terms (PDF) about listening to the “music of the primes,” but so far we’re still mostly in the dark about why primes are distributed the way they are, and what meta-patterns may still be waiting to be uncovered.
Prime numbers are pretty important in terms of cryptography; they’re what make RSA encryption worth the processor cycles used to run the numbers. This is because, while it’s easy to multiply two prime numbers together and get a huge number, it’s almost impossible to quickly go the other way ’round. We don’t have a simple method for teasing apart the prime factors of huge numbers; anyone who’s ever had calculus understands how completely the guess-and-check method is made of fail. So with regards to RSA, we’re currently embroiled in a long game of processing-power chicken. Finding out a clear, provable pattern leading to better insight into the distribution of primes could weaken the advantage of asymmetric cryptography like RSA encryption.
Patterns in mathematics, as in any other field, are often suspected long before they’re rigorously confirmed. Mathematical functions have an elegance to them, while biology is “messy,” often requiring slush factors and constants to bring our understanding of the rules into line with our observations. There are mathematical constants and entities everywhere in nature, though, from the fine-scale properties of molecular bonds to the prime-number life cycles of the cicada. Their presence demands we pay some attention to the nature of math itself: Are these laws and theorems something we’re discovering, or are they just an emergent property of our own attempts to quantize and describe our world? Nobel laureate physicist Eugene Wigner wrote in 1960, “The miracle of the appropriateness of the language of mathematics to the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”
Ultimately, while we don’t know this specific insight on patterns in prime numbers is reflected in the biosphere, or how to apply it to further illuminate mathematical theory, the fact that prime numbers are part of the living world makes it pretty clear they will factor into our future.
Featured image and data visualization are by SievesofChaos.