p-regularity of the p-adic valuation of the Fibonacci sequence

Details p-regularity of the p-adic valuation of the Fibonacci sequence p-regularity of the p-adic valuation of the Fibonacci sequence

2009-10-15

arxiv.org/abs/0910.2907v2

Mathematics

Number Theory

Scientific paper

This paper studies the $p$-adic valuation of the sequence $\{F_n\}_{n \geq 1}$ of Fibonacci numbers from the perspective of regular sequences. We establish that this sequence is $p$-regular for every prime $p$ and give an upper bound on the rank for primes such that Wall's question has a negative answer. We also point out that for primes $p \equiv 1,4 \mod 5$ the $p$-adic valuation of $F_n$ depends only on the $p$-adic valuation of $n$ and on the sum modulo $p-1$ of the base-$p$ digits of $n$ -- not the digits themselves or their order.

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