CONGRUENCES OF THE FIBONACCI NUMBERS MODULO A PRIME

Congruences of the form F(expr1) equivalent to epsilon F(expr2) (mod p) by prime modulo p are proved, whenever expr1 is a polynomial with respect to p. The value of epsilon 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), ..., a(2), a(1), a(0) and with respect to p of form 5t +/- 2; then, F(A(p)) equivalent to F(a(k) + a(k-1) +...+ a(2) + a(1) + a(0)) (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 congruences, experiments were performed in the Wolfram Mathematica system.