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Cycles in the generalized Fibonacci sequence modulo a prime. (English) Zbl 1076.11010
A generalized Fibonacci sequence $$\{G(n)\}$$ satisfies the recurrence: $G(n)= aG(n- 1)+ bG(n- 2),$ where $$a$$, $$b$$, $$G(0)$$, $$G(1)$$ are given integers. Assume that the associated characteristic equation: $$\lambda^2= a\lambda+ b$$ has distinct integer roots, so that $$a^2+ 4b= x^2> 0$$. The authors consider the behavior of $$\{G(n)\}\pmod p$$, where $$p$$ is prime. In particular, let $$C(p)= c$$ be the least positive integer such that $G(n+ c)\equiv G(n)\pmod p\text{ and }G(n+ c+ 1)\equiv G(n+1)\pmod p$ for all sufficiently large $$n$$. The authors show that if $$p$$ is a prime such that $$(x,p)= (b,p)= 1$$, then $$C(p)|(p- 1)$$.

##### MSC:
 11B39 Fibonacci and Lucas numbers and polynomials and generalizations
##### Keywords:
generalized Fibonacci sequence
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