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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Nonexistence conditions of a solution
for the congruence $x_1^k+\cdots+x_s^k\equiv N\protect\pmod{p^n}$


Authors: Hiroshi Sekigawa and Kenji Koyama
Journal: Math. Comp. 68 (1999), 1283-1297
MSC (1991): Primary 11D79; Secondary 11P05
Published electronically: February 24, 1999
MathSciNet review: 1627821
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Abstract | References | Similar Articles | Additional Information

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

DOI: http://dx.doi.org/10.1090/S0025-5718-99-01067-4
PII: S 0025-5718(99)01067-4
Keywords: Congruences, Diophantine equations, solvability, Waring's problem in $p$-adic fields, computer search
Received by editor(s): December 1, 1997
Published electronically: February 24, 1999
Article copyright: © Copyright 1999 American Mathematical Society