From Fermat to Gauss: An Examination of Congruences, Quadratic Reciprocity, and Binary Quadratic Forms

Abstract

Though the concepts of divisibility and remainders existed long before Gauss’s Disquisitiones Arithmeticae, it was not until then that the notion of a congruence was formally introduced. In this paper we return to some of Gauss’s fundamental ideas involving congruences. We inevitably encounter ideas by Fermat, Legrange, and Euler, among others, that are used in and lead into the main theorems of this paper. For example, we see Fermat’s Little Theorem useful in multiple of our proofs. We find in Theorem 10 at the end of chapter 1 that a number n = x 2 + y 2 , x, y ∈ Z if and only if the prime factors of n that are congruent to 3 mod 4 have an even exponent. We come to address the important Law of Quadratic Reciprocity at this paper’s summit in chapter 2 and finish with an exploration of binary quadratic forms, utilizing results from earlier in the paper involving congruences.