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Nonexistence Conditions of a Solution for the Congruence $x^k_1 + \cdots + x^k_s \equiv N (mod p^n)$

Hiroshi Sekigawa and Kenji Koyama
Mathematics of Computation
Vol. 68, No. 227 (Jul., 1999), pp. 1283-1297
Stable URL: http://www.jstor.org/stable/2584963
Page Count: 15
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Nonexistence Conditions of a Solution for the Congruence $x^k_1 + \cdots + x^k_s \equiv N (mod p^n)$
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Abstract

We obtain nonexistence conditions of a solution for of the congruence x$^k_1 + \cdots + x^k_s \equiv N (\mod p^n)$, where k ≥ 2, s ≥ 2 and N are integers, and pn is a prime power. We give nonexistence conditions of the form (s, N mod pn) for k = 2, 3, 4, 5, 7, and of the form (s, pn) 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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