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We obtain nonexistence conditions of a solution for of the congruence x k 1 +... + x k s ≡ N (mod p n ), where k ≥ 2, s ≥ 2 and N are integers, and p n is a prime power. We give nonexistence conditions of the form (s, N mod 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.

Nonexistence conditions of a solution for the congruence 𝑥₁^{𝑘}+⋯+𝑥_{𝑠}^{𝑘}≡𝑁(\mod𝑝ⁿ)