Although I had another blog post planned for today (also about primes surprisingly!), I read about an exciting new discovery on prime numbers in the news, and thought I’d write a quick blog post about it.

Kannan Soundararajan and Robert Lemke Oliver of Stanford University in the US have discovered a surprising pattern in the final digit of prime numbers. Apart from 2 and 5, all prime numbers have to end in 1, 3, 7 or 9 to prevent them from being divisible by 2 or 5. You would expect then that each of these 4 numbers have an equal probability (25%) of being the final digit of the next prime number, if the sequence of prime numbers is truly random.

However, the mathematicians made a discovery while performing a randomness check on the first 400 billion primes: prime numbers tend to avoid having the same number as their predecessor. In fact, it was found that:

- a prime ending in 1 is followed by another ending in 1 just 18.5% of the time (considerably lower than 25%);
- primes ending in 3 were more likely followed by primes ending in 9, rather than 1 or 7.

Although this isn’t necessarily a pattern, it does hint at the fact that the sequence of prime numbers isn’t completely random, something that mathematicians have taken to be true for many years. Ken Ono told Quanta Magazine:

*“I was floored. I have to rethink how I teach my class in analytic number theory now.”*

The two mathematicians suggest that the pattern they’ve discovered could be explained by the k-tuple conjecture, which predicts how groupings of primes will appear. Although this may not help in other prime related problems such as the twin-prime conjecture or the Riemann hypothesis, and may in fact have no implications to maths or number theory, Andrew Granville told the 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.”*

What do you think about the news? M x

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In case you wish to skim through the paper, can refer to this exposition by Terrence Tao: https://terrytao.wordpress.com/2016/03/14/biases-between-consecutive-primes/

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