An improved bound for prime solutions of some ternary equations (Q1081635)

Let \(b_ 1,b_ 2,b_ 3\) be non-zero integers, not all of the same sign, such that \((b_ j,b_ k)=1\) for \(j\neq k\). Let n be either 1 or 2 satisfying \(\sum ^{3}_{j=1}b_ j\equiv n (mod 2)\). In this paper we prove that there are positive absolute constants A and \(B_ 0\) such that the equation \[ b_ 1 p_ 1+b_ 2 p_ 2+b_ 3 p_ 3=n \] has prime solutions \(p_ 1,p_ 2,p_ 3\) satisfying max \(p_ j<B^ A\) where \(B=\max \{B_ 0, | b_ 1|, | b_ 2|, | b_ 3| \}.\) It is easy to see that the constant A cannot be replaced by 1. So, irrespective to the value of A, the above result gives the best possible bound for prime solutions of this type of equations. The original work on this problem can be traced back to a paper due to \textit{A. Baker} [J. Reine Angew. Math. 228, 166-181 (1967; Zbl 0155.092)].