Abstract

In 1996 Erdős conjectured that the set $\mathrm{\Sigma }(\mathrm{Pow}<!--FUNCTION\; APPLICATION-->(\{3,4\}),1)$ defined as the sums of distinct powers of 3 and distinct powers of 4 has positive asymptotic density. We investigate some structure properties of this set. We also prove some asymptotic estimates for its counting function $P_{\{3,4\}}(x)$. In particular we prove that $P_{\{3,4\}}(x)\gg x^{0.97777}$, improving an old estimate of Melfi.

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