Unexpected biases in the distribution of consecutive primes

Significance Prime numbers play a central role in analytic number theory, and are well known to be very well distributed among the reduced residue classes ( mod q ) . Surprisingly, the same does not appear to be true for sequences of consecutive primes, with different patterns occurring with wildly different frequencies. We formulate a precise conjecture, based on the Hardy−Littlewood conjectures, which explains this phenomenon. In particular, we predict that all patterns do occur their fair share of the time in the limit, but that there are secondary terms only very slowly tending to zero that create the observed biases.