A note on the completeness of an exponential type sequence

Abstract

For any given coprime integers p and q greater than 1, in 1959, B. J. Birch proved that all sufficiently large integers can be expressed as a sum of pairwise distinct terms of the form p ^a q ^b. As Davenport observed, Birch’s proof can be modified to show that the exponent b can be bounded in terms of p and q. In 2000, N. Hegyvari gave an effective version of this bound. The author improves this bound.

Keywords

Complete sequence / Coprime / Residue

References

[1] Birch B. J.. Note on a problem of Erdős. Proc. Cambridge Philos. Soc., 1959, 55: 370-373

[2] Hegyvari N.. On the completeness of an exponential type sequence. Acta Math. Hungar., 2000, 86(1–2): 127-135

[3] Vu V. H.. Some new results on subset sums. J. Number Theory, 2007, 124(1): 229-233