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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Nonexistence conditions of a solution for the congruence $x_1^k +\cdots + x_s^k \equiv N (\mod p^n)$
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by Hiroshi Sekigawa and Kenji Koyama PDF
Math. Comp. 68 (1999), 1283-1297 Request permission

Abstract:

We obtain nonexistence conditions of a solution for of the congruence $x_1^k+\cdots +x_s^k\equiv N\pmod {p^n}$, where $k\geq 2$, $s\geq 2$ and $N$ are integers, and $p^n$ is a prime power. We give nonexistence conditions of the form $(s, N\bmod {p^n})$ for $k=2$, $3$, $4$, $5$, $7$, and of the form $(s, p^n)$ for $k=11$, $13$, $17$, $19$. Furthermore, we complete some tables concerned with Waring’s problem in $p$-adic fields that were computed by Hardy and Littlewood.
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Additional Information
  • Hiroshi Sekigawa
  • Affiliation: NTT Communication Science Laboratories, 2-4 Hikaridai, Seika-cho, Soraku-gun Kyoto 619-0237 Japan
  • Email: sekigawa@cslab.kecl.ntt.co.jp
  • Kenji Koyama
  • Affiliation: NTT Communication Science Laboratories, 2-4 Hikaridai, Seika-cho, Soraku-gun Kyoto 619-0237 Japan
  • Email: koyama@cslab.kecl.ntt.co.jp
  • Received by editor(s): December 1, 1997
  • Published electronically: February 24, 1999
  • © Copyright 1999 American Mathematical Society
  • Journal: Math. Comp. 68 (1999), 1283-1297
  • MSC (1991): Primary 11D79; Secondary 11P05
  • DOI: https://doi.org/10.1090/S0025-5718-99-01067-4
  • MathSciNet review: 1627821