Prime Numbers Conspiracy

The press is full of reports about a new discovery about prime numbers.

Consecutive prime numbers do NOT like to end by the same digit mod q.

For example the probability that the difference of two consecutive prime numbers close to x is divisible by 3 is different than 1/2 one might very naively expect. It is substantially smaller and equal approximately to:



The term of conspiracy was used in Quanta Magazine paper and by the UK Independent and many others. We stress that this bias though very substantial is now apparently satisfactorily explained by mathematicians.
Mathematicians remain nevertheless stunned and shocked by this discovery. One frequently cited  expert on this and many similar questions is the UCL Maths department professor Andrew Granville. He told the press that this discovery is simply crazy or simply very surprising.

Is this Relevant to Cryptography?

This is a huge bias in practice. It is certainly relevant for prime numbers used in cryptography, can be used to accelerate some algorithms etc. It can be explained in terms of correlations: prime numbers lying close to a certain number x are correlated. This is quite disturbing and is very hard to believe.

Note: In cryptography we have lots of unexplained “mystery” questions, see for example at nearly every page of this paper or the “Mystery Question” section inside this more recent paper.



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