Divisibility properties of the Fibonacci entry point

Paul Cubre; Jeremy Rouse

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arXiv · 1212.6221

Divisibility properties of the Fibonacci entry point

Paul Cubre·Jeremy Rouse

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Abstract

For a prime $p$, let $Z(p)$ be the smallest positive integer $n$ so that $p$ divides $F_{n}$, the $n$th term in the Fibonacci sequence. Paul Bruckman and Peter Anderson conjectured a formula for $\zeta(m)$, the density of primes $p$ for which $m | Z(p)$ on the basis of numerical evidence. We prove Bruckman and Anderson's conjecture by studying the algebraic group $G : x^{2} - 5y^{2} = 1$ and relating $Z(p)$ to the order of $\alpha = (3/2,1/2) \in G(\F_{p})$. We are then able to use Galois theory and the Chebotarev density theorem to compute $\zeta(m)$.

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Paul Cubre, Jeremy Rouse. 2012-12-26. Divisibility properties of the Fibonacci entry point. https://arxiv.org/abs/1212.6221

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