Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-23T10:41:33.107Z Has data issue: false hasContentIssue false
  • English
  • Français

On a theorem in the additive theory of numbers due to evelyn and linfoot

Published online by Cambridge University Press:  24 October 2008

L. Mirsky
L. Mirsky
Affiliation:
Department of MathematicsUniversity of Sheffield
Get access Rights & Permissions [Opens in a new window]

Extract

Some years ago Evelyn and Linfoot found an asymptotic formula, with estimation of remainder, for the number of representations of a large integer n as the sum of s r-free integers. For s ≥ 4 their formula was subsequently sharpened by Barham and Estermann, and for s = 3 recently by me.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 44 , Issue 3 , July 1948 , pp. 305 - 312
Copyright
Copyright © Cambridge Philosophical Society 1948

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

* Evelyn, C. J. A. and Linfoot, E. H., ‘On a problem in the additive theory of numbers’, (I), Math. Z. 30 (1929), 433–48;Google Scholar(II) J. Math. 164 (1931), 131–40;Google Scholar(III) Math. Z. 34 (1932), 637–44;Google Scholar(IV), Ann. Math. 32 (1931), 261–70;Google Scholar(V), Quart. J. Math. 3 (1932), 152–60.Google Scholar

† If r ≥ 2, then an integer is called r-free if it is not divisible by the rth power of any prime.

‡ Barham, C. L. and Estermann, T., ‘On the representations of a number as the sum of four or more N-numbers’, Proc. London Math. Soc. (2), 38 (1935), 340–53.Google Scholar

§ Mirsky, L., ‘On the number of representations of an integer as the sum of three r-free int, egers’, Proc. Cambridge Phil. Soc. 43 (1947), 433–41.CrossRefGoogle Scholar I take this opportunity for correcting a misprint in that paper. The inequality in equation (15) on p. 440 should read

∥ Evelyn, C. J. A. and Linfoot, E. H., ‘On a problem in the additive theory of numbers’, (VI), Quart. J. Math. 4 (1933), 309–14.CrossRefGoogle Scholar

We may note at once that the problem is trivial if n≢sk (mod q), or if the highest common factor of q and k is not r-free, since in either case Q r, s(n, q, k) = 0.

* See, for example, Scholz, A., Einführung in die Zahlentheorie (Göschen), Satz31.Google Scholar