Conditional bounds for small prime solutions of linear equations

Abstract

Leta ₁,a ₂,a ₃ be non-zero integers with gcd(a ₁ a ₂,a ₃)=1 and letb be an arbitrary integer satisfying gcd (b, a _i,a _j) =1 fori≠j andb≡a ₁+a ₂+a ₃ (mod 2). In a previous paper [3] which completely settled a problem of A. Baker, the 2nd and 3rd authors proved that ifa ₁,a ₂,a ₃ are not all of the same sign, then the equationa ₁ p ₁+a ₂ p ₂+a ₃ p ₃=b has a solution in primesp _j satisfying

$$\mathop {\max }\limits_{1 \leqslant j \leqslant 3} p_j \leqslant 3\left| b \right| + (3\mathop {\max }\limits_{1 \leqslant j \leqslant 3} \left| {a_j } \right|)^A $$

whereA>0 is an absolute constant. In this paper, under the Generalized Riemann Hypothesis, the authors obtain a more precise bound for the solutionsp _j . In particular they obtainA<4+∈ for some ∈>0. An immediate consquence of the main result is that the Linnik's courtant is less than or equal to 2.