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A proof of quadratic reciprocity via linear recurrences

Volume 199 / 2021

Thomas Mack Acta Arithmetica 199 (2021), 433-440 MSC: Primary 11A15; Secondary 11B39, 11B37. DOI: 10.4064/aa210213-21-3 Published online: 28 June 2021


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].


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