>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, Granville and Ono agreed, 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).
This seems like an odd assumption to me. Surely sexy primes are more common than twin primes, so at least for primes that are near each other there should be a higher probability for certain sequences of final digits. This is obviously not proof in itself, but it would certainly make me hesitate to assume there is an equal probability in the general case.
> Surely sexy primes are more common than twin primes, so at least for primes that are near each other there should be a higher probability for certain sequences of final digits.
I think that's conjectural, but prime constellations are also conjectured to be a negligible fraction of primes as a whole, much as primes are a negligible fraction of integers (asymptotically). I think twin primes are conjectured to be distributed as n / (log n) ^2, while primes are n / (log n).
Besides, even if (p, p + 6) is significantly more common than (p, p + 2), if the final digit of p is uniformly distributed among 1, 3, 7, and 9, I don't think the statistics as a whole are affected.
This seems like an odd assumption to me. Surely sexy primes are more common than twin primes, so at least for primes that are near each other there should be a higher probability for certain sequences of final digits. This is obviously not proof in itself, but it would certainly make me hesitate to assume there is an equal probability in the general case.