Taiwanese Journal of Mathematics

FIBONACCI NUMBERS MODULO CUBES OF PRIMES

Zhi-Wei Sun

Full-text: Open access

Abstract

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

Article information

Source
Taiwanese J. Math., Volume 17, Number 5 (2013), 1523-1543.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706223

Digital Object Identifier
doi:10.11650/tjm.17.2013.2488

Mathematical Reviews number (MathSciNet)
MR3106028

Zentralblatt MATH identifier
1316.11013

Subjects
Primary: 11B39: Fibonacci and Lucas numbers and polynomials and generalizations 11B65: Binomial coefficients; factorials; $q$-identities [See also 05A10, 05A30]
Secondary: 05A10: Factorials, binomial coefficients, combinatorial functions [See also 11B65, 33Cxx] 11A07: Congruences; primitive roots; residue systems

Keywords
Fibonacci numbers central binomial coefficients congruences Lucas sequences

Citation

Sun, Zhi-Wei. FIBONACCI NUMBERS MODULO CUBES OF PRIMES. Taiwanese J. Math. 17 (2013), no. 5, 1523--1543. doi:10.11650/tjm.17.2013.2488. https://projecteuclid.org/euclid.twjm/1499706223


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