Summary: 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.