Japanese Journal of Mathematics

, Volume 2, Issue 2, pp 297–302

# On some number-theoretic conjectures of V. Arnold

Original Article

## 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.

### Keywords and phrases:

Euler congruences algebraic integers multiinomial coefficients

### Mathematics Subject Classification (2000):

05A10 11A07 11C20 11R04 11S15

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### References

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