Japanese Journal of Mathematics

, Volume 5, Issue 2, pp 183–189

# Generalizations of Arnold’s version of Euler’s theorem for matrices

Article

## Abstract.

A recent result, conjectured by Arnold and proved by Zarelua, states that for a prime number p, a positive integer k, and a square matrix A with integral entries one has $${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k})$$. We give a short proof of a more general result, which states that if the characteristic polynomials of two integral matrices AB are congruent modulo p k then the characteristic polynomials of A p and B p are congruent modulo p k+1, and then we show that Arnold’s conjecture follows from it easily. Using this result, we prove the following generalization of Euler’s theorem for any 2 × 2 integral matrix A: the characteristic polynomials of A Φ(n) and A Φ(n)-ϕ(n) are congruent modulo n. Here ϕ is the Euler function, $$\prod_{i=1}^{l} p_i^{\alpha_i}$$ is a prime factorization of n and $$\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2$$.

### Keywords and Phrases:

Euler congruences Euler’s theorem Fermat’s little theorem congruences for traces

### Mathematics Subject Classification (2010):

05A10 11A07 11C20

## Preview

### References

1. 1.
V.I. Arnold, Fermat–Euler dynamical systems and the statistics of arithmetics of geometric progressions, Funct. Anal. Appl., 37 (2003), 1–15.
2. 2.
V.I. Arnold, The topology of algebra: Combinatorics of squaring, Funct. Anal. Appl., 37 (2003), 177–190.
3. 3.
V.I. Arnold, Topology and statistics of formulae of arithmetics, Russian Math. Surveys, 58 (2003), 637–664.
4. 4.
V.I. Arnold, Fermat dynamics, matrix arithmetics, finite circles, and finite Lobachevsky planes, Funct. Anal. Appl., 38 (2004), 1–13.Google Scholar
5. 5.
V.I. Arnold, The matrix Euler–Fermat theorem, Izv. Math., 68 (2004), 1119–1128.Google Scholar
6. 6.
V.I. Arnold, Geometry and dynamics of Galois fields, Russian Math. Surveys, 59 (2004), 1029–1046.Google Scholar
7. 7.
V.I. Arnold, Ergodic and arithmetical properties of geometrical progression’s dynamics and of its orbits, Mosc. Math. J., 5 (2005), 5–22.Google Scholar
8. 8.
V.I. Arnold, On the matricial version of Fermat–Euler congruences, Jpn. J. Math., 1 (2006), 1–24.Google Scholar
9. 9.
W. Jänichen, Über die Verallgemeinerung einer Gaussschen Formel aus der Theorie der höhern Kongruenzen, Sitzungsber. Berlin. Math. Ges., 20 (1921), 23–29.Google Scholar
10. 10.
11. 11.
I. Schur, Arithmetische Eigenschaften der Potenzsummen einer algebraischen Gleichung, Compos. Math., 4 (1937), 432–444.
12. 12.
E.B. Vinberg, On some number-theoretic conjectures of V. Arnold, Jpn. J. Math., 2 (2007), 297–302.Google Scholar
13. 13.
A.V. Zarelua, On matrix analogs of Fermat’s little theorem, Math. Notes, 79 (2006), 783–796.
14. 14.
A.V. Zarelua, On congruences for the traces of powers of some matrices, Proc. Steklov Inst. Math., 263 (2008), 78–98.