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Pairs of additive congruences: cubic congruences
Author(s) -
Cook R. J.
Publication year - 1985
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300011062
Subject(s) - congruence relation , mathematics , prime (order theory) , pure mathematics , field (mathematics) , congruence (geometry) , combinatorics , geometry
We consider two additive cubic equationsa 1 x 1 3 + … + a n x n 3 = 0 ,b 1 x 1 3 + … + b n x n 3 = 0 ,in p ‐adic fields. Davenport and Lewis showed that the equations have a non‐trivial solution in every p ‐adic field, if n ≥ 16, and need not have a solution in the 7‐adic field, if n = 15. Here we prove that if p ≠ 7 the equations have a non‐trivial solution in p ‐adic fields if n ≥ 13. When n = 12 such a result fails for every prime p ≡ 1 (mod 3).