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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Sums of powers in large finite fields
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by Charles Small PDF
Proc. Amer. Math. Soc. 65 (1977), 35-36 Request permission

Abstract:

If k is a positive integer, then, in any finite field with more than ${(k - 1)^4}$ elements, every element is a sum of two kth powers.
References
    A. Hurwitz, Über die Kongruenz $a{x^e} + b{y^e} + c{z^e} \equiv 0\; \pmod p$, J. Reine Angew. Math. 136 (1909), 272-292.
  • Jean-René Joly, Équations et variétés algébriques sur un corps fini, Enseign. Math. (2) 19 (1973), 1–117 (French). MR 327723
  • I. Kaplansky, Private communications, October 7, 1975 and August 25, 1976.
  • Trygve Nagell, On the solvability of some congruences, Norske Vid. Selsk. Forh., Trondheim 27 (1954), no. 3, 5. MR 0064066
  • Th. Skolem, Unlösbarkeit von Gleichungen, deren entsprechende Kongruenz für jeden Modul lösbar ist, Avh. Norske Vid.-Akad. Oslo I 1942 (1942), no. 4, 28 (German). MR 16380
  • Charles Small, Waring’s problem $\textrm {mod}$ $n$, Amer. Math. Monthly 84 (1977), no. 1, 12–25. MR 424734, DOI 10.2307/2318299
  • Charles Small, Solution of Waring’s problem $\textrm {mod}\ n$, Amer. Math. Monthly 84 (1977), no. 5, 356–359. MR 439787, DOI 10.2307/2319964
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 65 (1977), 35-36
  • MSC: Primary 12C15
  • DOI: https://doi.org/10.1090/S0002-9939-1977-0485801-3
  • MathSciNet review: 0485801