A surprising pattern in the expression of prime numbers has been found by mathematicians, revealing a previously unknown “bias”.
Primes, as you’ll hopefully remember from school, are numbers that can only be divided by one or themselves (e.g. 2, 3, 5, 7, 11, 13, 17, etc). Their appearance in the roll call of all integers cannot be predicted, and no magical formula exists to know when a prime number will choose to suddenly make an appearance. It’s an open question as to whether or not a pattern even exists, or whether or not mathematicians will ever crack the code of primes, but most mathematicians agree that there’s a certain randomness to the distribution of prime numbers that appear back-to-back.
Or at least that’s what they thought. Recently, a pair of mathematicians decided to test this “randomness” assumption, and to their shock, they discovered that it doesn’t actually exist. As reported in New Scientist, researchers Kannan Soundararajan and Robert Lemke Oliver of Stanford University in California have detected unexpected biases in the distribution of consecutive primes.
The mathematicians made the discovery while performing a randomness check on the first hundred million primes. Within that set, a prime ending in 1 is followed by another ending in 1 18.5 per cent of the time. That shouldn’t happen if they were truly random — we would expect to see this happen 25 per cent of the time (keep in mind that primes can only end in 1, 3, 7, or 9). In terms of the back-to-back distribution of the other numbers, primes ending in 3 and 7 appeared 30 per cent of the time, and consecutive 9s appears about 22 of the time. Importantly, this observation has nothing to do with the base-10 numbering system, and is something inherent to primes themselves.
That’s a very odd discovery and certainly unexpected. So what’s going on? According to Soundararajan and Lemke Oliver, it may have something to do with what’s called the k-tuple conjecture, which is an old idea about how often pairs, triples, and larger sets of primes make an appearance.
As mathematician and founder of ClearerThinking.org explained to Gizmodo, the k-tuple conjecture is an attempt to understand the proximity of prime numbers to themselves. “Or more precisely, as we get larger and larger numbers, how often do we see prime numbers that have other primes nearby,” he said. “And it allows you to get very precise about what you mean by ‘nearby’.” For example, Greenberg says mathematicians can study primes that have a fixed spacing to five other primes. Basically, the k-tuple conjecture puts constraints on finding a prime that’s close to another prime. The new study has some interesting things to say about these constraints.
“As the numbers get larger, though, it sounds like this is less constraining, causing it to get closer and closer to an equal distribution of ending digits, which makes intuitive sense, since the primes get rarer and rarer,” he said.
Greenberg said it’s important to remember that primes, like the digits of pi, feel really random, but they’re not random at all. “They are determined precisely by the properties of numbers — it’s just that when we stare at their occurrence, our brains can’t see the pattern, so it feels like random madness.”
As fascinating as the new study appears to be, it likely won’t help with other prime-related problems, including the twin-prime conjecture or the Riemann hypothesis. And in fact, this discovery may have no practical use or implications to math and number theory. But as mathematician Andrew Granville told New Scientist, “It gives us more of an understanding, every little bit helps. If what you take for granted is wrong, that makes you rethink some other things you know.” [New Scientist, Nature News]