Congruent solubility of some additive equations (Q5917750)

In [Théorie des nombres, C. R. Conf. Int., Québec/Can. 1987, 595-624 (1989; Zbl 0682.10043)] and [Monatsh. Math. 111, 147-169 (1991; Zbl 0719.11064)] \textit{M.-C. Liu} and \textit{K.-M. Tsang} gave upper bounds for the smallest prime solution of the diophantine equation \(a_1 p^k_1+\cdots+ a_{s(k)} p^k_{s(k)}= b\), where \(k= 1\) or 2, \(s(1)= 3\), \(s(2)= 5\), \(b\) and the \(a_j\)'s are integers satisfying some necessary conditions, in terms of \(b\) and the \(a_j\)'s. In this paper, the author extends the above results to the cases \(k= 3\) and \(k= 4\), gives a sufficient condition for the existence of solutions of the above equation with \(s(3)= 9\) and \(s(4)= 17\), provided that \(b\) is large enough in terms of the \(a_j\)'s, and lower bounds for the number of solutions.