CONGRUENCES OF THE FIBONACCI NUMBERS MODULO A PRIME

Congruences of the form F(expr1) ≡ εF(expr2) (mod p) by prime modulo p are proved, whenever expr1 is a polynomial with respect to p. The value of ε 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 ak, ak–1, …, a2, a1, a0 and with respect to p of form 5t ± 2; then, F(A(p)) ≡ F(ak + ak–1 + … + a2 + a1 + a0) (mod p). In particular, we consider the case when the coefficients of the polynomial expr1 form the Pisano period modulo p. To search for existing сongruences, experiments were performed in the Wolfram Mathematica system.