A proof of quadratic reciprocity via linear recurrences
Volume 199 / 2021
Acta Arithmetica 199 (2021), 433-440
MSC: Primary 11A15; Secondary 11B39, 11B37.
DOI: 10.4064/aa210213-21-3
Published online: 28 June 2021
Abstract
In a 1999 preprint, A. Nakhash uses the recurrence relation for the Fibonacci numbers and their closed form over $\mathbb Q (\sqrt {5})$ to provide a proof of quadratic reciprocity specifically for the prime $5$. In this note, we construct a similar recurrence relation that extends this argument to arbitrary odd primes, answering an open problem in F. Lemmermeyer’s book [Reciprocity Laws, Springer, 2000, App. C].
