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 a very young age, while preparing for the International Mathematical Olympiad (see ^{[1][2][3][4][5][6][7]}).

The conjecture states the following:

For every prime number $p>2$ there exist two prime numbers $p_1,$ $p_2,$ with $p_1<p_2,$ 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 ^[4]).

Relation to other open problems

Rassias' conjecture, can be stated equivalently as follows:

For any prime number $p>2$ there exist two prime numbers $p_1,$ $p_2$ with $p_1<p_2,$ such that

$(p-1)p_1=p_2+1,$

namely the numbers $(p-1)p_1$ and $p_2$ are consecutive.

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

$p_2=2ap_1-1,$

strengthened by the additional condition that $2a+1$ be a prime number too (see ^[3][4]). 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 $p,$ $2ap+1\in\mathbb{P}$, where $\mathbb{P}$ denotes the set of prime numbers.

Probably the most general conjecture on distribution of prime constellations is Schinzel's hypothesis H:

Consider $s$ polynomials $f_{i}(x) \in \mathbb{Z}[X],\ i = 1, 2, \ldots, s$ with positive leading coefficients and such that the product $F(X) = \prod_{i=1}^{s} f_{i}(x)$ is not divisible, as a polynomial, by any integer different from ±1. Then there is at least one integer $x$ for which all the polynomials $f_{i}(x)$ take prime values.

Rassias' conjecture follows from the well-known Schinzel's hypothesis H for $s = 2$ with $f_{1}(x) = x$ and $f_{2}(x) = 2ax-1$. Note that Schinzel's hypothesis H appeared much earlier than Rassias' conjecture which is its special case. The reader is referred to the foreword of Preda Mihăilescu^[7] for a presentation of interconnections of Rassias' conjecture with other known conjectures and open problems in Number Theory. Further, another relevant open problem is related to Cunningham chains, i.e. sequences of primes

$p_{i+1}=mp_i+n,\ i=1,2,\ldots, k-1,$

for fixed coprime positive integers $m,n>1$.

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,^[8] there is no general result known on large Cunningham chains to date. Rassias' conjecture can be also stated in terms of Cunningham chains, namely that there exist Cunningham chains with parameters $2a, -1$ for $a$ such that $2a-1=p$ is a prime number (see ^[3][4]).

See also

• Sophie Germain prime
• Cunningham chain
• Green–Tao theorem

References

1. Andreescu, T.; Andrica, D. (2009). Number Theory: Structures, Examples and Problems. Birkhäuser, Boston, Basel. p. 12.
2. Balzarotti, G.; Lava, P. P. (2013). La Derivata Arithmetica. Editore Ulrico Hoepli Milano. pp. 140–141.
3. Mihăilescu, Preda (2011). "Book Review". Newsletter of the European Math. Soc. 79: 45–47.
4. Mihăilescu, Preda (2014). "On some conjectures in Additive Number Theory". Newsletter of the European Math. Soc. 92: 13–16.
5. Rassias, M. Th. (2005). "Open Problem No. 1825". Octogon Mathematical Magazine 13: 885.
6. Rassias, M. Th. (2007). "Problem 25". Newsletter of the European Math. Soc. 65: 47.
7. Rassias, M. Th. (2011). Problem-Solving and Selected Topics in Number Theory. Springer. pp. xi–xiii, 82.
8. Green, Ben; Tao, Terence (2008). "The primes contain arbitrarily long arithmetic progressions". Annals of Mathematics 2: 481–547. doi:10.4007/annals.2008.167.481.