# Two academics from Stanford University in California have shocked the world of mathematics by discovering a pattern in prime numbers, redefining the belief that prime numbers could be treated as if they occur randomly.

Making the discovery

Apart from 2 and 5, all prime numbers have to end in 1, 3, 7 or 9 so that they can’t be divided by 2 or 5. So if the numbers occurred randomly, as has been the long-held belief, then it wouldn’t matter what the last digit of the previous prime number was. Each of the four possibilities (1, 3, 7 or 9) should have an equal 25 % chance of appearing at the end of the next prime number.

Stanford mathematicians Kannan Soundararajan and Robert Lemke Oliver devised a computer programme to search for the first 400 billion prime numbers. Through this, they discovered that primes ending in 1 were less likely to be followed by another prime number ending in 1. This should not be the case if all prime numbers are truly random.

More specifically, a prime number ending in 1 was followed by another prime ending in 1 only 18.5 % of the time, significantly less than the expected 25 %. The two also discovered that prime numbers ending in 3 tended to be followed by primes ending in 9 more often than 1 or 7.

Explaining ‘the conspiracy among primes’

Soundararajan and Lemke Oliver though believe they have an explanation for the pattern, now being referred to as ‘the conspiracy among primes’.

Much of the modern research into prime numbers is underpinned by the work of early twentieth century Cambridge University mathematicians G. H. Hardy and John Littlewood. They devised a method to estimate how often pairs, triples and larger grouping of primes will appear, known as the ‘k-tuple conjecture’.

As Albert Einstein’s theory of relativity is an advance on Sir Isaac Newton’s theory of gravity, the k-tuple conjecture is essentially a more complicated version of the assumption that prime numbers are random. This finding demonstrates how the two assumptions differ.

The Stanford pair used Hardy and Littlewood’s work to highlight that the groupings given by the conjecture are responsible for introducing the last digit pattern, as they place restrictions on where the last digit of each prime can fall. However, the two also point out that as prime numbers stretch towards infinity, they eventually lose the last-digit pattern and start to appear in a much more random order.

Commenting to ‘New Scientist’ magazine, Prof Soundararajan commented: ‘It was very weird… it’s like some painting you are familiar with, and then suddenly you realise that there is a figure in the painting that you’ve never seen before.’

Moving forwards

The new result will not have any immediate applications to long-standing problems concerning prime numbers, such as the twin-prime conjecture (the assertion that there are infinitely many twin primes, or pairs of primes that differ by two digits, such as 3 and 5 or 5 and 7) or the Riemann hypothesis.

However, it has given the field a shock to its system. ‘I was floored… I have to rethink how I teach my class in analytic number theory now,’ said Ken Ono, a number theorist at Emory University in Atlanta, US to ‘Quanta Magazine’.