Did Euler Know Quadratic Reciprocity?: New Insights from a Forgotten Work - Conclusion - Appendix

Conclusion

A careful reading of Euler's De divisoribus numerorum in forma \(mxx + nyy\) contentorum (On divisors of numbers contained in the form \(mxx + nyy\)) (E744), then, shows that he did not make much progress along Edwards' ladder of theorems.  A statement of Lemma 5.1 does appear (in this more general setting) in Section 4 of the paper; nothing else indicates that Euler had any notion of reciprocity.  While we continue to be impressed with E744, we cannot from this work see any further evidence that Euler had an inkling of quadratic reciprocity in the sense in which we usually understand it.  The credit for this theorem must continue to rest with Euler's intellectual heirs, Legendre and Gauss.

Appendix

Editor's note: The authors have translated Euler's De divisoribus numerorum in forma \(mxx + nyy\) contentorum (E744) from Latin into English.  Read their translation of Euler's E744, On divisors of numbers contained in the form \(mxx + nyy.\)

About the Authors

Paul Bialek is an Associate Professor in the Department of Mathematics at Trinity International University in Deerfield, Illinois.  His mathematical interests include the history of mathematics, number theory and the Swiss mathematician Leonhard Euler, especially the translation of Euler’s works from Latin to English.  Besides Latin, he has studied Spanish, New Testament Greek, Old Testament Hebrew, Chinese, French and Russian.  Paul is a member of the MAA, the Euler Society, and the Association of Christians in the Mathematical Sciences, and he is frequently a Reader for the AP Calculus Exam.

Dominic Klyve is an Associate Professor of Mathematics at Central Washington University, where he also directs the Math Honors Program.  Due to the sometimes unfortunate fact that he finds many different things interesting, he works in Number Theory, the History of Mathematics, and Applied Statistics.  He serves on the Executive Board of the Euler Society, the Editorial Board of the College Mathematics Journal, and is a Councilor on the Council on Undergraduate Research.  Dominic enjoys spending time with his family, and spends quite a bit of his non-work life playing with stuffed animals and Legos®.