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Mathematicians find a peculiar pattern in primes

By sciencenews.org

Prime numbers, divisible only by 1 and themselves, hate to repeat themselves. They prefer not to mimic the final digit of the preceding prime, mathematicians have discovered.

“It is really, really bizarre,” says Stanford University postdoctoral researcher Robert Lemke Oliver, who, with Stanford number theorist Kannan Soundararajan, discovered this unusual prime predilection. “We are still trying to understand what is at the heart of this,” Lemke Oliver says.

Generally speaking, primes are thought to behave much like random numbers. So whenever some kind of order is revealed, it gives mathematicians pause.

“Any regularity you can show about primes is beguiling, because there may lurk there some new structure,” says number theorist Barry Mazur of Harvard University. “Revealing some bit of architecture where we thought there was none may lead to inroads into the structure of the mathematics.”

Once primes get into the double digits, they must end in either a 1, 3, 7 or 9. Mathematicians have long known that there are roughly the same number of primes ending with each digit; each appears as the final number about 25 percent of the time. The prime number theorem in arithmetic progressions proved this distribution about 100 years ago, and the still unsolved Riemann hypothesis predicts that the rates rapidly approach 25 percent. This property has been tested for many millions of primes, says Soundararajan.

And without any reason to think otherwise, mathematicians just assumed that the distribution of those final digits was basically random. So given a prime that ends in 1, the odds that the next prime ends in 1, 3, 7 or 9 should be roughly equal….

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