On some number-theoretic conjectures of V. Arnold

E.B. Vinberg

Abstract: In [1], V.I. Arnold conjectured "the matrix Euler congruence": $\text{tr} A^{p^n}\equiv\text{tr} A^{p^{n-1}} \ (\text{mod}\ p^n)$ for any integer matrix $A$, prime $p$, and natural number $n$. He proved it for $p\le 5$, $n\le 4$. 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.