On the completeness of an exponential type sequence (Q1586345)

Confirming a conjecture of P. Erdős, in 1959 \textit{B. J. Birch} [Proc. Camb. Philos. Soc. 55, 370-373 (1959; Zbl 0093.05003)] proved that for any coprime integers \(p, q>1\) every sufficiently large integer is a sum of distinct numbers of the form \(p^\alpha q^\beta \). The author shows that summands with the restriction \(\beta \leq K\) suffice for a suitable \(K=K(p,q)\), and exhibits such a bound \(K\). The bound is quadruply exponential in \(p\) and triply in \(q\), probably not the true order of magnitude.