Discovery of a Pattern in Prime Numbers Stuns Mathematicians

March 18, 2016 | Joanne Kennell


They are not so random after all!

Two mathematicians stunned themselves and the world of mathematics by discovering a pattern in prime numbers. It turns out, primes are actually a bit picky about who their neighbors are.

Prime numbers are those that are divisible only by themselves and 1, but they are also the building blocks of all other numbers which are created by multiplying primes together. Cleary, prime numbers are fundamental to arithmetic and this is why mathematicians are constantly trying to decipher their mysteries.

Currently, mathematicians do not have a way of predicting which numbers are prime, so they have been treated as if they occur randomly. However, Kannan Soundararajan and Robert Lemke Oliver of Stanford University in California have discovered that the previous way of thinking needs to change a little.

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“It was very weird,” said Soundararajan to New Scientist. “It’s like some painting you are very familiar with, and then suddenly you realise there is a figure in the painting you’ve never seen before.”

So what was the big surprise? Aside from 2 and 5, all prime numbers end in 1, 3, 7 or 9 since they can’t be divided by 2 or 5. If the numbers occurred randomly as expected, it shouldn’t matter what the last digit of the previous prime was since each of the four possibilities should have a 25 percent chance of appearing at the end of the next prime number.

It turns out, that is not what’s happening. While searching through the first hundred million primes, the two mathematicians noticed that primes ending in 1 were followed by another ending in 1 just 18.5 percent of the time — something that would not happen if the primes were truly random. Primes ending in 3 and 7 each followed 1 a whopping 30 percent of the time, while 9 followed a 1 in 22 percent of occurrences.

Similar patterns appeared for other combinations of endings, all deviating from the expected random values, and the numbers really tended to avoid having the same last digit as their immediate predecessor. Meaning, they hate to repeat themselves.

The patterns become more random as you count higher — the team developed a computer programme to search through the first 400 billion primes — but the pattern still persisted.

“In ignorance, we thought things would be roughly equal,” Andrew Granville of the University of Montreal, Canada told New Scientist. “One certainly believed that in a question like this we had a very strong understanding of what was going on.”

So where does this pattern come from? Soundararajan and Lemke Oliver think they have an explanation — the k-tuple conjecture. This conjecture was developed by G. H. Hardy and John Littlewood, mathematicians from the University in Cambridge in the early 20th century. It is a way of estimating how often pairs, triples and larger groupings of primes appear.

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Using Hardy and Littlewood’s work, the team showed that the groupings given by the conjecture were responsible for the last-digit pattern. However, they also found that as the primes stretched to infinity, they completely lost the pattern and followed the random distribution mathematicians are used to expecting.

The k-tuple conjecture still needs to be proven, but many mathematicians believe it is correct because it has been extremely useful in predicting the behavior of primes. “It is the most accurate conjecture we have, it passes every single test with flying colours,” said James Maynard of the University of Oxford, UK, to New Scientist. “If anything I view this result as even more confirmation of the k-tuple conjecture.”

Although the discovery won’t have any immediate implications in the world of mathematics, it did leave some mathematicians a little shaken. “It gives us more of an understanding, every little bit helps,” said Granville to New Scientist. “If what you take for granted is wrong, that makes you rethink some other things you know.”

The results were published in the online journal arXiv.org.

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