On some number-theoretic conjectures of V. Arnold

摘要

In [1], V.I. Arnold conjectured “the matrix Euler congruence” tr Apn º tr Apn-1 (textmod pn){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.