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The solubility of certain Diophantine inequalities

Published online by Cambridge University Press:  26 February 2010

H. Davenport  and
K. F. Roth
H. Davenport
Affiliation:
University College, London
K. F. Roth
Affiliation:
University College, London
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Extract

The method devised by Hardy and Littlewood for the solution of Waring's Problem, and further developed by Vinogradov, applies quite generally to Diophantine equations of an additive type. One particular result that can be proved by this method is that if a1, …, as are integers not all of the same sign, and if s is greater than a certain number depending only on k, the Diophantine equation

has infinitely many solutions in positive integers x1, …, xs, provided that the corresponding congruence has a non-zero solution to every prime power modulus. A result of a similar kind holds if on the right of (1) there stands an arbitrary integer in place of 0.

Type
Research Article
Information
Mathematika , Volume 2 , Issue 2 , December 1955 , pp. 81 - 96
Copyright
Copyright © University College London 1955

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References

page 81 note * Journal London Math. Soc., 21 (1946), 185–193.Google Scholar

page 82 note * See The method of trigonometrical sums in the theory of numbers (Interscience Publishers, 1954), Ch. VI.Google Scholar

page 82 note † (a) Acta Math., 71 (1939), 123–143CrossRefGoogle Scholar; (b) Annals of Math., 40 (1939), 731–747CrossRefGoogle Scholar; (c) American J. of Math., 64 (1942), 199–207.CrossRefGoogle Scholar

page 83 note * We use Vinogradov's abbreviation as an equivalent for F = O(G).

page 84 note * See, for example, Lemma 6 of Ch. II of Vinogradov's book.

page 87 note * The value of the integral is of course real, and the same applies to similar integrals later.

page 90 note * Quart. J. of Math. (Oxford), 9 (1938), 199–202.Google Scholar

page 94 note * Perron, , Die Lehre von den Kettenbrüchen (Leipzig and Berlin, 1929), Satz, 16 of §15 combined with inequality (12) of §13.Google Scholar