On the matricial version of Fermat-Euler congruences
The paper discusses some generalizations of the Little Fermat Theorem and its generalization to general \(n\) due to Euler. To be more specific in the two statements \(a^p \equiv a \bmod p\) and \(a^n \equiv a^{n-\varphi(n)} \bmod n\) where \(\varphi(n)\) is the Euler totient function, the \(a\) is replaced by the trace of an integer matrix \(A\). The matricial version of the second statement turns out to be wrong for \(n \neq p^b\) where \(p\) is prime, but seems to hold otherwise. The stated congruences can be translated in particular congruences involving symmetric functions. The congruences on the symmetric functions can be shown to hold if certain congruences between particular multinomial coefficients hold. These congruences are shown to hold (by an inductive argument) provided some basic (starting) conditions hold. These starting conditions have to be verified for every set of parameters separately (infinitely many). In the paper the author verifies starting conditions for \(p=3\) and \(p=5\) and some parameter \(r=3\), and claims that a computer verified them for all primes less than 30. This then also would prove the matricial Euler congruences mod \(p^r\) \(p= 3\) or 5 and \(r\) less than 5. On the fly Arnold introduces what he calls Newton-Girard formulas expressing \(x_1^n+\dots+x_r^n\) in terms of the moments of the elementary symmetric functions \(\sigma_1=x_1+\dots +x_r\), \(\sigma_2=x_1x_2+\dots\), \(\sigma_r=x_1x_2\cdots x_r\), that were already proved in an earlier paper of the same author. For checking the initial conditions on the multinomial coefficients the author found some useful shortcuts reducing the amount of work considerably. The paper does not prove the presented matricial form of the Euler congruences but rather indicates a way to prove it for separate sets of parameters. Finally the paper connects the presented material to some general number theoretic problems like the behaviour of the minimal period of the geometric progression of residues mod \(n\) and the behaviour of the mean divisor of \(n\). The presented material is interesting and deserves further study. In reading the paper I found the notation used by the author rather peculiar and I had the feeling that much could be gained by using a decent notation, e.g. for multinomial coefficients. Much of the material then can be presented in a clearer and more accessible way. Later on, studying a paper by \textit{E. B. Vinberg} [``On some number-theoretic conjectures of V. Arnold'', Jpn. J. Math. (3) 2, No. 2, 297--302 (2007; Zbl 1168.05004)], my last remark turned out to be true. By introducing the appropriate notation and reformulation of the statements in the proper way, Vinberg is able to prove the matricial Euler congruences that are conjectured and disproves some conjectures on multinomial coefficients in this paper (and to replace them by the best possible (proved) alternative).
scientific article; zbMATH DE number 5503380