MIND-BOGGLING

# Mathematicians are geeking out about a bizarre discovery in prime numbers

Prime numbers have both fascinated and boggled mathematicians for millennia. But a new study contends that one aspect of prime numbers’ core usefulness—the ability to appear random—may not be what we suspected it to be.

Despite having a simple definition, a whole number that cannot be exactly divided by any other number than 1 or itself, prime numbers display many levels of complexity. They don’t, for instance, show a discernible pattern, and thus appear random. It is this uniqueness that is used to encrypt important information swapped over the open internet. For instance, every time you make an online purchase with a credit card, prime numbers spring into action to complete that transaction for you securely.

Researchers studying prime numbers at Stanford University have stumbled upon a new phenomenon. In their study, published on arXiv, they show that consecutive prime numbers try hard not to be similar. That is, they may not be as random as once thought.

Apart from the single-digit prime numbers 2 and 5, all other prime numbers can only end in one of four digits: 1, 3, 7, or 9. (If a number ends in 2, 4, 6, 8 or 0, it will be divisible by 2. If it ends in 5, it will be divisible by 5.) Thus, if they were truly random, a prime number that ends in 1 should be followed by another prime number ending in 1 about 25% of the time. That is, this kind of pairing should occur at least one in four times.

But when Kannan Soundararajan and Robert Lemke Oliver checked the first billion prime numbers, they found that a prime number ending in 1 is followed by another also ending in 1 about 18% of the time. That is, this kind of pairing occurred only one in five times. Instead, that prime number was followed by a prime number ending in 3 or 7 about 30% of the time and by 9 about 22% of the time. This result holds true for prime numbers ending in 3, 7 or 9 too, but with slightly less bias.

Their study is yet to be checked by experts before it is accepted for publication in a peer-reviewed journal, but mathematicians are geeking out about the results already. “Every single person we’ve told this ends up writing their own computer program to check it for themselves,” Soundararajan told Nature.

As with anything to do with numbers, this bizarre pattern has always existed. The researchers only found it now because they went looking for it. Fortunately, the “anti-sameness” bias doesn’t yet have any practical implication on the rules of cryptography that underpin our important online transactions. But mathematicians are happy to be stumped. They have a new challenge to explain the phenomenon. As we know from historical examples, this hardy group of scholars won’t remain stumped for long.