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.
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References
Birch, B. J., Note on a problem of Erdős, Proc. Cambridge Philos. Soc., 55, 1959, 370–373.
Hegyvari, N., On the completeness of an exponential type sequence, Acta Math. Hungar., 86(1–2), 2000, 127–135.
Vu, V. H., Some new results on subset sums, J. Number Theory, 124(1), 2007, 229–233.
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Project supported by the National Natural Science Foundation of China (Nos. 10771103, 11071121).
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Fang, J. A note on the completeness of an exponential type sequence. Chin. Ann. Math. Ser. B 32, 527–532 (2011). https://doi.org/10.1007/s11401-011-0660-5
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DOI: https://doi.org/10.1007/s11401-011-0660-5