Among the first billion prime numbers, for instance, a prime ending in 9 is almost 65 percent more likely to be followed by a prime ending in 1 than another prime ending in 9. In a paper posted online today [March 13, 2016], Kannan Soundararajan and Robert Lemke Oliver of Stanford University present both numerical and theoretical evidence that prime numbers repel other would-be primes that end in the same digit, and have varied predilections for being followed by primes ending in the other possible final digits…

This conspiracy among prime numbers seems, at first glance, to violate a longstanding assumption in number theory: that prime numbers behave much like random numbers. Most mathematicians would have assumed… that a prime should have an equal chance of being followed by a prime ending in 1, 3, 7 or 9 (the four possible endings for all prime numbers except 2 and 5).

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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One thought on “A curious non-randomness in the distribution of primes”

I suspect, though I haven’t read the paper, that the primes do exhibit such a bias despite being as random as they can be, given initial conditions. Note that the 8 co-primes of 30, namely (7, 11, 13, 17, 19, 23, 29, 31), create 8 arithmetic series of the form c + 30k, k a non-negative integer, for a particular coprime c. Each of these 8 series contains an infinite number of primes and account for all primes larger than 5. Looking at the series ending in 9 as 19+30k and 29+30k, the average gap to the next possible prime ending in 9, is given by 22.5, or (10+30+20+30)/4, whereas the average gap from a 9 prime to the next higher possible prime ending in 1, i.e., 31+30k and 41+30k, is 12, or (2+12+12+22). This might be enough to skew the results as described in the paper, at least for the first million primes. I would expect the described effect would dampen out as one gets farther north and the incidence of primality continues to decrease and the front-end effect is lessened in the average.

I suspect, though I haven’t read the paper, that the primes do exhibit such a bias despite being as random as they can be, given initial conditions. Note that the 8 co-primes of 30, namely (7, 11, 13, 17, 19, 23, 29, 31), create 8 arithmetic series of the form c + 30k, k a non-negative integer, for a particular coprime c. Each of these 8 series contains an infinite number of primes and account for all primes larger than 5. Looking at the series ending in 9 as 19+30k and 29+30k, the average gap to the next possible prime ending in 9, is given by 22.5, or (10+30+20+30)/4, whereas the average gap from a 9 prime to the next higher possible prime ending in 1, i.e., 31+30k and 41+30k, is 12, or (2+12+12+22). This might be enough to skew the results as described in the paper, at least for the first million primes. I would expect the described effect would dampen out as one gets farther north and the incidence of primality continues to decrease and the front-end effect is lessened in the average.