A clear rule determines exactly what makes a prime: it’s a whole number that can’t be exactly divided by anything except 1 and itself. But there’s no discernable pattern in the occurrence of the primes. Beyond the obvious — after the numbers 2 and 5, primes can’t be even or end in 5 — there seems to be little structure that can help to predict where the next prime will occur.
As a result, number theorists find it useful to treat the primes as a ‘pseudorandom’ sequence, as if it were created by a random-number generator.
- Article pdf: UNEXPECTED BIASES IN THE DISTRIBUTION OF CONSECUTIVE PRIMES
: http://arxiv.org/pdf/1603.03720v4.pdfMaths whizz solves a master’s riddle
But if the sequence were truly random, then a prime with 1 as its last digit should be followed by another prime ending in 1 one-quarter of the time. That’s because after the number 5, there are only four possibilities — 1, 3, 7 and 9 — for prime last digits. And these are, on average, equally represented among all primes, according to a theorem proved around the end of the nineteenth century, one of the results that underpin much of our understanding of the distribution of prime numbers. (Another is the prime number theorem, which quantifies how much rarer the primes become as numbers get larger.)
Instead, Lemke Oliver and Soundararajan saw that in the first billion primes, a 1 is followed by a 1 about 18% of the time, by a 3 or a 7 each 30% of the time, and by a 9 22% of the time. They found similar results when they started with primes that ended in 3, 7 or 9: variation, but with repeated last digits the least common. The bias persists but slowly decreases as numbers get larger.The k-tuple conjecture
The mathematicians were able to show that the pattern they saw holds true for all primes, if a widely accepted but unproven statement called the Hardy–Littlewood k-tuple conjecture is correct. This describes the distributions of pairs, triples and larger prime clusters more precisely than the basic assumption that the primes are evenly distributed.
The idea behind it is that there are some configurations of primes that can’t occur, and that this makes other clusters more likely. For example, consecutive numbers cannot both be prime — one of them is always an even number. So if the number n is prime, it is slightly more likely that n + 2 will be prime than random chance would suggest. The k-tuple conjecture quantifies this observation in a general statement that applies to all kinds of prime clusters. And by playing with the conjecture, the researchers show how it implies that repeated final digits are rarer than chance would suggest.
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