Let \(k\geq 2\) be an integer and \(s=2^k+1\) for \(k\leq 11\), \(s=2[k^2(2\log k+\log \log k+2.6)]+1\) if \(k\geq 12\). Let \(b_1,\ldots,b_s\) be any non-zero integers which are not all of the same sign, \((b_1,\ldots,b_s)=1\) and satisfy the condition of congruent solubility. Then there exists a constant \(A>0\) such that the equation \(b_1p^k_1+\ldots+b_sp^k_s=0\) has a solution in primes \(p_1,\ldots,p_s\) satisfying \(\max_{1\leq j\leq s}p_j\leq(2\prod^s_{j=1}| b_ j|)^A\).
The case \(k=1\) and \(s=3\) was first considered by A. Baker [J. Reine Angew. Math. 228, 166–181 (1967; Zbl 0155.09202)] and was completely settled by K.-M. Tsang and the reviewer [Théorie des Nombres, C. R. Conf. Int., Québec/Can. 1987, 595–624 (1989; Zbl 0682.10043)]. For the case \(k=2\) and \(s=5\), essentially the same result as the above was obtained also by K.-M. Tsang and the reviewer [Monatsh. Math. 111, 147–169 (1991; Zbl 0719.11064)].
The method used in the proof is the Hardy-Littlewood method and the above best possible bound for \(p_ j\) is obtained mainly due to a result of P. X. Gallagher [Invent. Math. 11, 329–339 (1970; Zbl 0219.10048)] on the distribution of zeros of \(L\)-functions.