ON THE GREATEST PRIME DIVISOR OF THE SUM OF TWO SQUARES OF PRIMES

Abstract

One of the most famous theorems in number theory states that there are infinitely many positive prime numbers (namely p = 2 and the primes p ≡ 1 mod4) that can be represented in the form x²₁+x²₂, where x₁ and x₂ are positive integers. In a recent paper, Fouvry and Iwaniec [2] have shown that this statement remains valid even if one of the variables, say x₂, is restricted to prime values only. In the sequel, the letter p, possibly with an index, is reserved to denote a positive prime number. As p²₁+p²₂ = p is even for p₁, p₂ > 2, it is reasonable to conjecture that the equation p²₁+p²₂ = 2p has an infinity of solutions. However, a proof of this statement currently seems far beyond reach. As an intermediate step in this direction, one may quantify the problem by asking what can be said about lower bounds for the greatest prime divisor, say P(N), of the numbers p²₁+p²₂, where p₁, p₂ [les ] N, as a function of the real parameter N [ges ] 1. The well-known Chebychev–Hooley method combined with the Barban–Davenport–Halberstam theorem almost immediately leads to the bound P(N) [ges ] N^1−ε, if N [ges ] N_o(ε); here, ε denotes some arbitrarily small fixed positive real number. The first estimate going beyond the exponent 1 has been achieved recently by Dartyge [1, Théorème 1], who showed that P(N) [ges ] N^10/9−ε. Note that Dartyge's proof provides the more general result that for any irreducible binary form f of degree d [ges ] 2 with integer coefficients the greatest prime divisor of the numbers [mid ]f(p₁, p₂)[mid ], p₁, p₂ [les ] N, exceeds N^γ_d−ε, where γ_d = 2 − 8/(d + 7). We in particular want to point out that Dartyge does not make use of the specific features provided by the form x²₁+x²₂. By taking advantage of some special properties of this binary form, we are able to improve upon the exponent γ₂ = 10/9 considerably.