On sums of distinct powers of 3 and 4
Résumé
In 1996 Erdős conjectured that the set Σ(Pow({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) ≫ x^0.97777, improving an old estimate of Melfi.