Rassias conjecture

In Number Theory, Rassias' conjecture (named after Michael Th. Rassias) is an open problem related to prime numbers. It was conceived by M. Th. Rassias at the age of 14, while preparing for the International Mathematical Olympiad (see , , , , , , ).

The conjecture states the following:

For any prime number p > 2, there exist two prime numbers p₁, p₂, with p₁ < p₂, such that

$$p=\frac{p_1+p_2+1}{p_1}.$$ This conjecture has a surprising feature of expressing a prime number as a quotient (see ).

Relation to other open problems

Rassias' conjecture, can be stated equivalently as follows:

For any prime number p > 2, there exist two prime numbers p₁, p₂, with p₁ < p₂, such that

(p−1)p₁ = p₂ + 1, namely the numbers (p−1)p₁ and p₂ are consecutive.

By this reformulation, we see an interesting combination of a generalized Sophie Germain twin problem

p₂ = 2ap₁ − 1, strengthened by the additional condition that 2a + 1 be a prime number too (see , ). We have seen that such questions are caught by the Hardy-Littlewood conjecture. One may ask if Rassias' conjecture is to some extent simpler than the general Hardy-Littlewood conjecture or its special case concerning distribution of generalized Sophie-Germain pairs p, 2ap + 1 ∈ ℙ, where ℙ denotes the set of prime numbers. This is not likely to be the case (see ).

Another relevant open problem, is related to Cunningham chains, i.e. sequences of primes

p_i + 1 = mp_i + n, i = 1, 2, …, k − 1, for fixed coprime positive integers m, n > 1.

There are computing competitions for the longest Cunningham chain or for the one built up of the largest primes - but unlike the breakthrough of Ben J. Green and Terence Tao, there is no general result known on large Cunningham chains to date. Rassias' conjecture can be also stated in terms of Cunningham chains, namely: there exist Cunningham chains with parameters 2a,  − 1 for a such that 2a − 1 = p is a prime number (see , ).

See also

• Sophie Germain prime
• Cunningham chain
• Green-Tao Theorem