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.