2024, September 27-28 ORESME Reading Group Meeting

Our 47th meeting will be held on Friday-Saturday, September 27-28, 2024, at Xavier University. This will be the second (and concluding) meeting that 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 are using 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 3-6 (and possibly 7) of this chapter (pp. 261-306). Following up on the work we did at our previous meeting in February of this year on the early history of the subject (Euclid's, Fermat's and Euler's contributions), we turn to its further elaboration and development of the symmetrical nature of this "reciprocity" in higher arithmetic through the work of Joseph-Louis Lagrange (1736--1813), Adrien-Marie Legendre (1752--1833), Carl Friedrich Gauss (1777--1855) and Gotthold Eisenstein (1823--1852). Available here are the same materials we shared for our previous meeting in February: a pdf of Chapter 4 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. 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.