# On some number-theoretic conjectures of V. Arnold

Author: Vinberg, E.

Source: Japanese Journal of Mathematics, Volume 2, Number 2, September 2007 , pp. 297-302(6)

Publisher: Springer

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Abstract:

In [1], V.I. Arnold conjectured “the matrix Euler congruence” $${\rm tr} A^{p^n}\equiv {\rm tr} A^{p^{n-1}}\,(\text{mod}\,{p^{n}})$$ for any integer matrix A, prime p, and natural number n. He proved it for p ≤ 5, n ≤ 4. In fact the conjecture immediately follows from a result of C.J. Smyth [5]. We give a simple proof of this result and discuss a related conjecture of Arnold concerning some congruences for multinomial coefficients.

Document Type: Research Article

Affiliations: Department of Mechanics and Mathematics, Moscow State University, Moscow, 119992, GSP–2, Russia, Email: vinberg@zebra.ru

Publication date: 2007-09-01

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