Quadratic forms representing the $p$-th Fibonacci number

TL;DR

This paper proves that for prime numbers p congruent to 1 or -1 modulo 4, the Fibonacci number F_p can be expressed as a quadratic form involving p, expanding understanding of Fibonacci number representations.

Contribution

It establishes new quadratic form representations for Fibonacci numbers at primes congruent to 1 or -1 mod 4, extending previous results.

Findings

F_p = u^2 - p v^2 for primes p ≡ 1 mod 4

F_p = u^2 + p v^2 for primes p ≡ -1 mod 4

Provides explicit quadratic form representations for these Fibonacci numbers

Abstract

In this paper, we show that if $p\equiv 1\pmod 4$ is prime, then $4F_p$ admits a representation of the form $u^2-pv^2$ for some integers $u$ and $v$, where $F_n$ is the $n$th Fibonacci number. We prove a similar result when $p\equiv -1\pmod 4$.