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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 \(C(p)|(p- 1)\).

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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