Rassias' conjecture

In number theory, Rassias' conjecture is an open problem related to prime numbers. It was conjectured by Michael Th. Rassias while preparing for the International Mathematical Olympiad. The conjecture states the following:

For every prime $p>2$ there exist two primes $p_{1}<p_{2}$ such that

$p={\frac {p_{1}+p_{2}+1}{p_{1}}}$

Relation to other open problems

Rassias' conjecture can be stated equivalently as follows:

For any prime number $p>2$ there exist primes $p_{1}<p_{2}$ such that

$p_{2}=(p-1)p_{1}-1.$

This reformulation shows that the conjecture is a combination of a generalized Sophie Germain prime problem

$p_{2}=2ap_{1}-1$

strengthened by the additional condition that $2a+1$ be prime too. This makes it a special case of Dickson's conjecture. Note that Dickson's conjecture (and its generalization, Schinzel's hypothesis H) appeared much earlier than Rassias' conjecture. See the foreword of Preda Mihăilescu for a presentation of interconnections of Rassias' conjecture with other known conjectures and open problems in number theory.

Also related are Cunningham chains, i.e. sequences of primes $p_{i+1}=mp_{i}+n,\ i=1,2,\ldots ,k-1,$ for fixed coprime positive integers $m,n>1$. Unlike the breakthrough of Ben Green and Terence Tao on primes in arithmetic progression, there is no general result known on large Cunningham chains to date. Rassias' conjecture is equivalent to the existence of Cunningham chains with parameters $2a,-1$ for $a$ such that $2a-1=p$ is prime.

References

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