Congruences of the Fibonacci numbers modulo a prime

V. M. Zyuz'kov

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 69 (2021), pp. 15-21

Résumé

Congruences of the form $F(expr1) \equiv\varepsilon F(expr2) \pmod p$ by prime modulo $p$ are proved, whenever $expr1$ is a polynomial with respect to $p$. The value of $\varepsilon$ equals $1$ or $-1$ and $expr2$ does not contain $p$. An example of such a theorem is as follows: given a polynomial $A(p)$ with integer coefficients $a_{k}, a_{k-1}, \dots , a_{2}, a_{1}, a_{0}$ and with respect to $p$ of form $5t \pm 1$; then, $F(A(p))\equiv F(a_{k} + a_{k-1} + \dots + a_{2} + a_{1} + a_{0}) \pmod p$. In particular, we consider the case when the coefficients of the polynomial expr1 form the Pisano period modulo $p$. To search for existing congruences, experiments were performed in the Wolfram Mathematica system.

Détail

Keywords: Fibonacci numbers, rangruences modulo a prime number, Pisano period, Mathematica system.