Geodesic
The matrix Euler--Fermat theorem
Izvestiya. Mathematics , Tome 68 (2004) no. 6, pp. 1119-1128.

Voir la notice de l'article provenant de la source Math-Net.Ru

We prove many 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. We thus have an extension of the matrix Fermat theorem similar to Euler's extension of the numerical little Fermat theorem. 
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V. I. Arnol'd. The matrix Euler--Fermat theorem. Izvestiya. Mathematics , Tome 68 (2004) no. 6, pp. 1119-1128. http://geodesic.mathdoc.fr/item/IM2_2004_68_6_a1/

[1] Arnold V. I., “Dinamika Ferma, arifmetika matrits, konechnaya okruzhnost i konechnaya ploskost Lobachevskogo”, Funktsion. analiz i ego prilozh., 38:1 (2004), 1–15 | MR | Zbl

[2] Arnold Problems, Phasis–Springer, Heidelberg–Berlin–N.Y, 2004, P. 157–162

[3] Girard A., Sur des decouvertes nouvelles en algèbre, Amsterdam, 1629

[4] Newton I., Arithmetica universalis, Cambridge, 1707