On Legendre's Work on the Law of Quadratic Reciprocity

Legendre was the first to state the law of quadratic reciprocity in the form that we know it and he was able to prove it in some but not all cases, with the first complete proof being given by Gauss. In this paper we trace the evolution of Legendre’s work on quadratic reciprocity in his four great works on number theory. As is well known, Adrien-Marie Legendre (1752-1833) was the first to state the law of quadratic reciprocity in the form that we know it (though an equivalent result had earlier been conjectured by Euler), and he was able to prove it in some but not all cases, with the first complete proof being given by Gauss [3]. In this paper we trace the evolution of Legendre’s work on quadratic reciprocity in his four great works [10, 11, 12, 13] on number theory. These works span a 45 year period in Legendre’s life, dating from 1785, 1797, 1808, and 1830 respectively. Before beginning with our analysis here, we call the reader’s attention to several other relevant works. [15] overlaps with our work here, though it has a somewhat different focus. [2] is a brief survey, and [14] is an extended treatment of the early history of reciprocity laws. A highly readable account of the development of number theory around this era can be found in [9], which has excerpts from original works of Euler, Legendre, Gauss, and others, translated into English. In this paper, we will use Legendre’s language to the extent possible. In particular, we will not use the terms quadratic residue/nonresidue or the notion of congruence in the body of this article, as these were never used by Legendre. We begin with Legendre’s 1785 paper [10]. In Article I of that paper he proves a result due originally to Euler: Theorem A. Let c be an odd prime and let d be any integer not divisible by c. Then d − 1 is divisible by c. Furthermore, c divides the formula x + dy (i.e., there are integers x and y not divisible by c with x + dy divisible by c) if and only if (−d) leaves a remainder of 1 when divided by c; otherwise (−d) leaves a remainder of −1 when divided by c. If −c/2 < d < c/2, each possibility occurs for (c − 1)/2 values of d. 1 Note that “c divides the formula x+dy” if and only if −d is a quadratic residue (mod c).