2024, February 8-9, ORESME Reading Group Meeting

Our 46th meeting will be held on Friday-Saturday, February 8-9, 2024, at Northern Kentucky University. This will be the first of two meetings we devote to the genesis of the Law of Quadratic Reciprocity, the amazing fact that relates whether the prime p is a quadratic residue mod q to whether q is a quadratic residue mod p.

To guide our study, we will use the excellent summary of this story in Chapter 4 ("Patterns in Prime Numbers: the Quadratic Reciprocity Law") of Mathematical Masterpieces by Knoebel, et al. (Springer, 2007).

At this meeting, we plan to follow sections 1-3 of this chapter (pp. 229-279). The authors of Mathematical Masterpieces present an overview of the story and focus on the contributions of Leonhard Euler (1707--1783)---of course, he must be involved!---and Joseph-Louis Lagrange (1736--1813). Available here is a pdf of the relevant chapter of Mathematical Masterpieces, Latin originals of both of Euler's papers and Lagrange's as well, together with translations of the first of Euler's papers and Lagrange's. But Mathematical Masterpieces includes translations of all the relevant passages we will need.

Here are the readings:

• Knoebel A, et al., Mathematical Masterpieces: Further chronicles by the explorers. Undergraduate Texts in Mathematics, Springer-Verlag. 2007.
• Leonhard Euler, Theoremata circa divisores numerorum in hac forma paa±qbb (1747). E164. Commentarii academiae scientiarum Petropolitanae, Volume 14 (1751), pp. 151-181. Also in Opera Omnia, Series 1, Volume 2, pp.194-222.
• ---, Theorems about the divisors of numbers contained in the form paa±qbb. E164. Translated by Jordan Bell, 2006. From the Euler Archive.
• ---, Observationes circa divisionem quadratorum per numeros primos (1772). E552. Opuscula analytica, Volume 1 (1783), pp. 64-84.
• Joseph-Louis Lagrange, Recherches d'Arithmétique (1773 & 1775). Nouveaux Mémoires de l'Académie royale des Science et Belle-lettres, 1773, pp. 265-312, 1775, pp. 325-356.
• ---, Research on Arithmetic. Translated by Erik Tou, 2021. From arXiv.