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Did Euler Know Quadratic Reciprocity?: New Insights from a Forgotten Work - A Capstone to a Life in Number Theory

Paul Bialek (Trinity International University) and Dominic W. Klyve (Central Washington University)

The paper De divisoribus numerorum in forma \(mxx + nyy\) contentorum (E744—On divisors of numbers contained in the form \(mxx + nyy\)), in which Euler returned for the last time to the questions he had raised 25 years earlier in his much better known E164, is quite interesting.  It contains an exciting new generalization of the types of quadratic forms Euler had studied in E164.  Also, Euler knew when writing it that it would not be published until long after his death—he had already submitted so many papers to the St. Petersburg Academy journal that he estimated at the time it would take 20 years for them to be to published [1].  This turned out to be an underestimate; Euler submitted E744 in 1778, and it was not published until 1815—a 37-year delay which makes even the slowest modern journal look comparatively sprightly.  Because of this, he had no reason to hold back any of his thoughts on the subject, and no fear that half-formed ideas could be picked up by others.  If, in E744, we can find evidence of the theorems given on the preceding page, we might see this as evidence that Euler was indeed moving toward quadratic reciprocity, and side with Edwards on Euler’s priority.  If even at this late date we can find no such evidence, we should see this as evidence to side with Sandifer, who contended that full understanding of quadratic reciprocity would have to await Gauss.

Despite the fact that E744 concerns Euler's final work on factors of quadratic forms, it seems never to have been seriously studied or written about.  We know of no secondary work which so much as references it.  It remains a largely unknown work, and our hope is that our new translation could shed some light on important questions about Euler's number theory.

Euler seems to have written E744 in 1778, kicking off a three-year period in which he returned with gusto to the study of Diophantine equations.  In this paper, he extended his earlier work on primes dividing numbers of the form \(x^2 + Ny^2\) to the more general question of primes dividing numbers of the form \(mx^2 + ny^2.\)  The paper, like so many of Euler's, is a masterwork of exposition.  Assuming that his readers may not have been familiar with his earlier work, he introduced the topic with straightforward examples and computations, building up to a general theorem.  His primary result, somewhat surprisingly, was that the primes which divide \(mx^2 + ny^2\) are determined solely by the product \(mn.\)            

Yet in some sense, E744 is disappointing.  By almost any measure, Euler was by 1778 the most famous and accomplished mathematician in Europe.  One of his (many!) projects over the previous half century had been establishing number theory, previously thought of as nothing more than a series of recreational puzzles, as a serious mathematical discipline.  Like many of his lifelong projects (cf. his plan for mechanics as described in [16, pp. 332–333]), this had been largely successful—although there is no evidence in this case that his number theory plan had been as intentional from the beginning.  Joseph-Louis Lagrange and Gauss were able to pick up Euler's tools and refine them into a body of knowledge which closely resembles the content of modern textbooks in number theory, establishing vocabulary, notation, and methods which remain in use.  A reader might hope that, 36 years after his first foray into the topic of factors of quadratic forms, Euler would have a new proof technique to share.  Instead, he repeated his earlier pattern, simply generating tables and stating “theorems” without proof.

It is quite possible that Euler sought this more general setting in an attempt to prove the results which he had been able only to state in his earlier works.  If so, the paper was a failure.  Seen on its own merits, however, full of computations, examples, and the deep insights into the nature of numbers that seemed to come so naturally to Euler, E744 is a successful simplification and generalization of the work he did earlier in such papers as E164 [4], E241 [5], and E256 [6].  For our purposes, we are primarily concerned with whether this paper contains any hint that Euler was thinking about quadratic reciprocity. 

The short answer is that it seems he was not.  Indeed, as far as we know, the very concept never occurred to him.  Rather, we see in E744 that Euler was concerned with a topic which was of interest to him for most of his working life—identifying the factors of quadratic forms.

Read the authors' translation of Euler's E744.

Paul Bialek (Trinity International University) and Dominic W. Klyve (Central Washington University), "Did Euler Know Quadratic Reciprocity?: New Insights from a Forgotten Work - A Capstone to a Life in Number Theory," Convergence (February 2014)