## Mathematicians Shocked to Find Pattern in ‘Random’ Prime Numbers

### Jacob Aron | March 19, 2016

[Blogger's note: Prime numbers form the basis for encryption algorithms. Seeing patterns in random consecutive primes raises unexpected questions. Are there implications for encryption algorithms?]
[...] Apart from 2 and 5, all prime numbers end in 1, 3, 7 or 9 (anything ending in 2 or 5 is divisible by that same number) – and each of the four endings is equally likely. But while searching through the primes, Soundararajan and Lemke Oliver noticed that primes ending in 1 are less likely to be followed by another ending in 1 than primes ending in 3, 7 or 9. That shouldn’t happen if primes are truly random – consecutive primes shouldn’t care about their neighbour’s final digit.The pair found that in the first hundred million primes, a prime ending in 1 is followed by another ending in 1 just 18.5 per cent of the time. If they were random, you’d expect to see two primes ending in 1 next to each other 25 per cent of the time. [...]

 Prime numbers up to 200. The green numbers are primes, the black are composite. Source: http://statistics.about.com/od/ProbHelpandTutorials/a/What-Is-The-Probability-Of-Randomly-Choosing-A-Prime-Number.htm

<more at https://www.newscientist.com/article/2081034-mathematicians-shocked-to-find-pattern-in-random-prime-numbers/; related links and articles: http://www.techtimes.com/articles/141021/20160315/mathematicians-discover-pattern-in-random-prime-numbers.htm (Mathematicians Discover Pattern In 'Random' Prime Numbers) and http://arxiv.org/abs/1603.03720 (Unexpected biases in the distribution of consecutive primes. Robert J. Lemke Oliver, Kannan Soundararajan. arXiv:1603.03720v2 [math.NT]. [Abstract: While the sequence of primes is very well distributed in the reduced residue classes (mod q), the distribution of pairs of consecutive primes among the permissible ϕ(q)2 pairs of reduced residue classes (mod q) is surprisingly erratic. This paper proposes a conjectural explanation for this phenomenon, based on the Hardy-Littlewood conjectures. The conjectures are then compared to numerical data, and the observed fit is very good.])>