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The matrix Euler-Fermat theorem. (English. Russian original) Zbl 1167.11300
Izv. Math. 68, No. 6, 1119-1128 (2004); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 68, No. 6, 61-70 (2004).
This is a continuation of the author’s article [Funct. Anal. Appl. 38, No. 1, 1–13 (2004); translation from Funkts. Anal. Prilozh. 38, No. 1, 1–15 (2004; Zbl 1125.11066)] where congruences generalizing Fermat’s little theorem are proved for the traces of powers of integer matrices. Here the author proves several congruences for binomial and multinomial coefficients as well as for the coefficients of the Girard-Newton formula in the theory of symmetric functions. These congruences also imply congruences (modulo powers of primes) for the traces of various powers of matrices with integer elements. Thus he has an extension of the matrix Fermat theorem similar to Euler’s extension of the numerical little Fermat theorem.

MSC:
11A07 Congruences; primitive roots; residue systems
11B50 Sequences (mod \(m\))
11C20 Matrices, determinants in number theory
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