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On a theorem of Birch concerning sums of distinct integers taken from certain sequences

Published online by Cambridge University Press:  28 June 2011

U. Zannier
U. Zannier
Affiliation:
Istituto di Matematica, Università di Salerno, Baronissi, Italy
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Extract

In [1] B. J. Birch, solving in the affirmative a conjecture of Erdὅs, proved the following result:

Theorem 1. Let p and q be coprime integers greater than 1. Then every large natural number may be written as a sum of distinct terms of type paqb.

In fact Birch pointed out that, with similar arguments, one could obtain a stronger version where the exponent b of q can be bounded in terms of p and q. The proofs were entirely elementary.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 106 , Issue 2 , September 1989 , pp. 199 - 206
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

[1] Birch, B. J.. Note on a problem of Erdὅs, Proc. Cambridge Soc. 55 (1959), 370–373.CrossRefGoogle Scholar
[2] Cassels, J. W. S.. On the representation of integers as the sums of distinct summands taken from a fixed set. Acta Sci. Math. (Szeged) 21 (1960), 111–124.Google Scholar
[3] Erdὅs, P.. On the representation of large integers as sums of distinct summands taken from a fixed set. Acta Arith. 7 (1962), 345–354.CrossRefGoogle Scholar
[4] Perelli, A. and Zannier, U.. On sums of distinct integers belonging to certain sequences. Acta Math. Hungar. 41 (1983), 251–254.CrossRefGoogle Scholar
[5] Zannier, U.. An elementary proof of some results concerning sums of distinct terms from a given sequence of integers. (To appear.)Google Scholar