Recurrence Relations over the Residue Field and Generalizations of the Markoff Equation

In this paper we analyse the classical diophantine Markov equation and its generalisations. It is proved that the Markov graph constructed on the basis of recurrence relations for the classical equation is a tree. The question of connectivity of the Markov graph for a generalised equation modulo a prime number p is investigated and the possibility of applying the recurrence approach to prove the existence of a large connectivity component is demonstrated. Next, we study the properties of the Pisano period for the Fibonacci sequence modulo natural number. A number of lemmas are proved which allow us to reduce the problem of period length estimation to the case of modulus equal to the degree of a prime number. Finally, a result on the number of points on a curve over a finite field is applied to estimate the length of the Pisano period for a prime modulo.