Fibonacci numbers modulo cubes of primes

Abstract

Let $p$ be an odd prime. It is well known that $F_{p-(\frac{p}{5})}\equiv 0(\mathrm{mod}p)$, where $\{F_n{\}}_{n\ge 0}$ is the Fibonacci sequence and $(-)$ is the Jacobi symbol. In this paper we show that if $p\ne 5$ then we may determine $F_{p-(\frac{p}{5})}$ mod $p^3$ in the following way: $$\sum _{k=0}^{(p-1)/2}\frac{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}{(-16{)}^k}\equiv \left(\frac{p}{5}\right)\left(1+\frac{F_{p-(\frac{p}{5})}}{2}\right)(\mathrm{mod}{p}^3).$$ We also use Lucas quotients to determine ${\sum }_{k=0}^{(p-1)/2}\left(\genfrac{}{}{0ex}{}{2k}{k}\right)/m^k$ modulo $p^2$ for any integer $m\not\equiv 0(\mathrm{mod}p)$; in particular, we obtain $$\sum _{k=0}^{(p-1)/2}\frac{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}{16^k}\equiv \left(\frac{3}{p}\right)(\mathrm{mod}{p}^2).$$ In addition, we pose three conjectures for further research.

21 pages. Final published version

References in corpus (4)

• On congruences related to central binomial coefficients
• On sums of binomial coefficients modulo p^2
• p-adic valuations of some sums of multinomial coefficients
• Proof of three conjectures on congruences

Cited by in corpus (4)

• Open Conjectures on Congruences
• On congruences related to central binomial coefficients
• Congruences involving binomial coefficients and Lucas sequences
• Curious congruences for Fibonacci numbers