A bound for prime solutions of some ternary equations
In the course of his paper [On some diophantine inequalities involving primes, J. Reine Angew. Math. 228, 166-181 (1967; Zbl 0155.092)], \textit{A. Baker} proved using Hardy-Littlewood-Vinogradov techniques that given any non-zero integers \(b_ 1,b_ 2,b_ 3\) with \((b_ 1,b_ 2,b_ 3)=1\) and which do not have the same sign, then for any \(\delta >0\), there exists a constant \(C=C(\delta)>0\) such that the equation \[ b_ 1p_ 1+b_ 2p_ 2+b_ 3p_ 3=m, \] where \(m=1\) or 2, has a solution is primes \(p_ 1,p_ 2,p_ 3\) with max \(p_ j<C^{\max | b_ j|^{\delta}}\). By avoiding a major arcs estimate which relies on the Siegel-Walfisz theorem, the author obtains a smaller bound when \(b_ 1,b_ 2,b_ 3\) are pairwise coprime and the integer m satisfies \(b_ 1+b_ 2+b_ 3\equiv m (mod 2)\) and \(1\leq m\leq (\log \max | b_ j|)+2,\) by showing that there exists an absolute and effectively computable positive constant A with \(\max p_ j\leq A^{(\log \max b_ j)^ 2+1}.\)
scientific article; zbMATH DE number 3891473
scientific article