Mathematicians are often interested in determining who was the first person to discover or prove an important mathematical concept or theorem. The more important the mathematics involved, and the more important the contenders, the more interesting the question. By this standard, the origin of Quadratic Reciprocity is one of the more interesting open questions in the history of number theory. 

Leonhard Euler (1707-1783) (Source: Convergence Portrait Gallery)

The question of who first knew quadratic reciprocity has been written about in the past.  Harold Edwards discussed the subject in 1983 [3].  After a careful study of one of Euler's most important papers on quadratic forms and an earlier letter to Christian Goldbach, Edwards came down on the side of Euler.  Twenty-two years later, Ed Sandifer took up the question again, and disagreed, ceding credit to Carl Friedrich Gauss [15]. 

Part of the debate may rest on one's answer to the question “what does it really mean to know a mathematical theorem?”  For the time being, we suggest the following tentative answer:  Mathematician \(M\) knows theorem \(T\) if, when someone else states theorem \(T\) to her, she can reply “yes---I already knew that.”  In this light we can make our question a bit more precise: if Euler had (magically) come across a modern statement of quadratic reciprocity, would he have nodded, unsurprised?  Or would he have learned something new?

For most such questions in number theory, we would immediately turn to Leonard Dickson's History of the Theory of Numbers [2].  Written in the early twentieth century, Dickson's work stands as the definitive compilation of the history of number theory before his time.  There is, however, one significant topic missing from his history: quadratic reciprocity.  Dickson's exclusion of quadratic reciprocity is an interesting story in itself (see [9]); for our purposes, we simply note that in the absence of a definitive history, we shall have to work a bit harder to come to our own conclusions.

In their respective papers, both Sandifer and Edwards cited a letter from Euler to Goldbach and Theoremata circa divisores numerorum in hac forma \(paa \pm qbb\) contentorum (E164) [4], Euler's best-known and most important work on the subject.  Over the rest of his career, however, Euler wrote several papers on related topics.  One of these (E744) [7] appears to have been totally neglected by historians.  We have recently completed a translation of E744, and in this paper, we use the new information contained therein about Euler's number theory near the end of his life to contribute to the debate about Euler and quadratic reciprocity.

Read the authors' translation of Euler's E744.

It is no exaggeration to call the Quadratic Reciprocity Theorem one of the most important in number theory.  The theorem describes when a prime \(p\) “is a square” modulo another prime \(q.\)  The simplest nontrivial example concerns the prime \(q=7.\)  Modulo \(7,\) it turns out that \(p=2\) is a square, because

\[3^2 = 9 \equiv 2\,{\rm{(mod}}\,7);\]

in this case, \(2\) is called a quadratic residue \({\rm{(mod}}\,7).\)  There are several equivalent formulations of quadratic reciprocity; one of the clearest is due to Adrien-Marie Legendre [13]:

Theorem 2.1.  Let \(p\) and \(q\) be odd primes.  Then:

1. If either \(p \equiv 1\,{\rm{(mod}}\,4)\) or \(q \equiv 1\,{\rm{(mod}}\,4),\) then \(p\) is a quadratic residue \({\rm{(mod}}\,q)\) if and only \(q\) is a quadratic residue \({\rm{(mod}}\,p).\)
2. If both \(p \equiv 3\,{\rm{(mod}}\,4)\) and \(q \equiv 3\,{\rm{(mod}}\,4),\) then \(p\) is a quadratic residue \({\rm{(mod}}\,q)\) if and only if \(q\) is not a quadratic residue \({\rm{(mod}}\,p).\)

Legendre published the first proof of this theorem in his Essai sur la théorie des nombres in 1798 [13].  Gauss had apparently found a proof in 1796 (see [3]), although this wasn't published until 1801, in his Disquisitiones Arithmeticae.  Gauss was very fond of this theorem—he eventually gave six different proofs!  Efforts to extend the theorem to higher degrees (e.g., cubic reciprocity) and to rings beyond the integers formed a crucial part of the development of number theory in the nineteenth century.  In fact, David Hilbert's ninth problem, to “Find the most general law of the reciprocity theorem in any algebraic number field,” reflects that even at the dawn of the 20th century, reciprocity continued to vex number theorists.  (A part of this story can be found in [17, pp. 163–257].)

A complete summary of Euler's work in number theory is far beyond the scope of the present article.  He wrote almost 100 papers on the subject, covering a wide range of elementary and analytic topics.  Here we will content ourselves to outline his work which relates to quadratic reciprocity. 

Early in his career, Euler became interested in the work of Pierre de Fermat.  This proved an excellent motivation for Euler, as Fermat stated interesting theorems, but rarely proved them.  One of Fermat's “theorems” (he never proved it), first stated in a letter to Marin Mersenne [10, II, 212–217], was that a prime which is one more than a multiple of 4 is a sum of two squares in one and only one way (see [14] for this and other early references to similar claims).  Euler gave a stronger version of Fermat’s claim in a 1742 letter to Goldbach, when he stated (in modern notation):

“Theorem” 3.1.  For \(x\) and \(y\) relatively prime, all prime divisors of \(x^2 + y^2\) are congruent to \(1\,{\rm{(mod}}\,4).\)  Furthermore, any prime \(p\) congruent to \(1\,{\rm{(mod}}\,4)\) divides \(x^2 + y^2\) for some \(x\) and \(y.\)

(Euler’s original letter can be found in [8].  Edwards' useful translation, from which we took our statement of “Theorem” 3.1, is in [3].)  In his letter to Goldbach, Euler claimed that this result was well known, but then extended it considerably, stating similar results for other quadratic forms.  He asserted, for example, that for relatively prime integers \(x\) and \(y\):

• All prime divisors of \(2x^2 + y^2\) are congruent to \(1\) or \(3\,{\rm{(mod}}\,8).\)  Furthermore, any prime \(p\) congruent to 1 or \(3\,{\rm{(mod}}\,8)\) divides \(2x^2 + y^2\) for some \(x\) and \(y.\)
• All prime divisors of \(3x^2 + y^2\) are congruent to \(1\) or \(7\,{\rm{(mod}}\,{12}).\)  Furthermore, any prime \(p\) congruent to 1 or \(7\,{\rm{(mod}}\,{12})\) divides \(3x^2 + y^2\) for some \(x\) and \(y.\)
• All prime divisors of \(5x^2 + y^2\) are congruent to \(1, 3, 5,\) or \(7\,{\rm{(mod}}\,{20}).\)  Furthermore, any prime \(p\) congruent to 1, 3, 5, or \(7\,{\rm{(mod}}\,{20})\) divides \(5x^2 + y^2\) for some \(x\) and \(y.\)

(Each of the bulleted points are taken from [3].)

As he admitted to Goldbach in 1742, Euler couldn't prove any of these statements at the time.  He would prove Theorem 3.1 in 1750 [5] (though the paper wouldn't be published for ten more years), allowing us to remove the quotation marks from its name.  He proved the first of our bulleted statements, concerning the form \(2x^2 + y^2,\) in 1753—this proof would appear in a delightful 1761 paper on the use of numerical experimentation in mathematics [6].

Euler's inability to prove these statements, however, did not stop him from stating a more general version of these ideas.  Euler claimed that, in general, the prime divisors of \(a^2 + Nb^2\) would all be congruent to \({s_i}\,{\rm{(mod}}\,{4N})\) for some \(s_i\) in a set \(S = \{s_1, s_2, \ldots, s_n \}\) depending on \(N.\)  Although Euler didn't give explicit construction rules for the set \(S,\) he did state several facts about it, notably that \(S\) is closed under multiplication \({\rm{(mod}}\,{4N}),\) and that for all \(x < 4N\) with \(\gcd(x,4N)=1,\) either \(x\) or \(4N-x\) (but not both) is in \(S.\)  When Euler instead considered quadratic forms \(a^2 - Nb^2,\) he had to modify this last claim.  The resulting statement is crucial to our present discussion, and we shall restate it more formally:

Claim 3.1 (Euler, from E164, Note 14).  All prime divisors of \(a^2 - Nb^2\) are congruent to some \({s_i}\,{\rm{(mod}}\,{4N}).\)  Furthermore, any prime \(p\) congruent to some \({s_i}\,{\rm{(mod}}\,{4N})\) divides \(a^2 - Nb^2\) for some \(a\) and \(b.\)  Conversely, these \(s_i\) all come from the set \(S,\) with the property that exactly half of the positive integers less than \(|2N|\) (and relatively prime to \(2N)\) are in \(S,\) and for all \(x <|2N|,\) \(x \in S\) if and only if \(|4N| -x \not\in S.\)

We have now come to the crux of the matter: when Euler considered this theorem, did he understand the statement of quadratic reciprocity?  Is this theorem equivalent to quadratic reciprocity?

How do we judge if a mathematician from a different time and place “knew” a statement or theory from modern mathematics?  Not only the language and notation, but even the standards of proof and description have changed dramatically over the centuries.  Euler's Claim 3.1 was presented by Euler as a “theorem,” as were dozens of other statements in E164, despite the fact that he provided no proofs.  Some people claim that Gauss knew group theory, since the elements of so much of modern group theory can be seen in his Disquisitiones.  Others claim that Fermat (or even Archimedes!) knew a body of theory equivalent to basic integral calculus.  We must not fall into the trap of claiming that somebody didn't know a modern result just because their vocabulary or standard of proof was different, but it is also dangerous to give credit where none is due, by anachronistically reading modern understanding into the words of historical figures (see the discussion in [12]).

What then shall we do?  With historical sensitivity, we must try to understand the work of the players involved.  We must be willing to translate their work into modern language (difficult to avoid when we are comparing their conceptions with ours), while being careful not to put words (or symbols) into the mouths (or pens) of the original writers.  We must be willing to read between the lines of what they wrote, but not to insert whole new lines of thought.  With this is mind, we turn to our big question: “Did Euler know quadratic reciprocity?”

We must first point out that neither Edwards nor Sandifer claimed that Euler directly stated the modern form of the quadratic reciprocity theorem.  Furthermore, Edwards argued that Euler's statement in Claim 3.1 is better than quadratic reciprocity in many ways, since the work of the next century showed that the most natural way to describe the general relationship between the quadratic character of residues for \(p\) and \(q\) is not to use “reciprocity” (that is, don't let \(p\) depend on \(q,\) as in Legendre’s statement above), but to give a description similar to Euler's.  Indeed, Edwards pointed out that the most general form of reciprocity, as given by Emil Artin's Reciprocity Law, contains no reciprocity at all!

This, however, requires us to peer deeply into Euler's future.  It seems that at the time when Euler was writing, it was more natural to think about reciprocity—explaining why we see it in the work of Legendre and Gauss.  How close, then, is Euler's statement to the standard form of quadratic reciprocity, as expressed in Theorem 2.1? 

Edwards, in his article, included a discussion about how one would prove a more standard version of quadratic reciprocity from Euler's Claim 3.1.  Extracting the major steps from his discussion, we find that there are three transitional steps through what we will call one lemma and two theorems.

Lemma 5.1.  For any integer \(N,\) odd squares are necessarily on its list \(S.\) That is, for any odd integer \(x,\) \(x^2 {\rm{(mod}}\,{4N})\) is in \(S.\)

Theorem 5.2.  The numbers which are in the set \(S\) for both \(N\) and \(-N,\) when \(N\) is prime, are precisely those numbers \(s,\) \(-2N < s < 2N,\) that can be written in the form \(s = t^2 - 4Nk\) where \(t\) is a positive odd integer less than \(N.\)

Theorem 5.3 (Transition to QR).  Theorem 5.2 implies quadratic reciprocity (Theorem 2.1).

Edwards pointed out that the lemma is obvious (we already know that the set \(S\) is closed under multiplication), and gave proofs of the two theorems above. 

We would argue that if these steps are “obvious,” then it probably is best to say that Euler knew quadratic reciprocity.  If, however, they are not, or if their proofs use tools Euler didn't have at his disposal, then we would do better to say that he did not. 

The reader is encouraged to try to prove these statements for herself.  Trying to judge the difficulty of a proof by its length is dangerous, but it may be worth noting that Edwards' proofs run about sixteen lines of small print in footnotes of his article.  The proofs are not deep, but in our opinion neither are these claims “obvious” in any way.

In summary, to go from his published work to quadratic reciprocity, Euler would need Theorems 5.2 and 5.3.  If these are simple enough that Euler might have known them or could have seen them trivially, we can give Euler credit.  Otherwise, we cannot.  We claim that the statements are not trivial, and Euler gave no indication in his early work that he was thinking about these statements.  There is, however, another way we can get insight into this question.

Near the end of his life, Euler revisited the topic of factors of quadratic forms.  In De divisoribus numerorum in forma \(mxx + nyy\) contentorum (E744—On divisors of numbers contained in the form \(mxx + nyy\)), Euler took up the questions we've seen throughout this article in a more general context.

Read the authors' translation of Euler's E744.

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.

We next describe what precisely Euler did include in the paper, De divisoribus numerorum in forma \(mxx + nyy\) contentorum (On divisors of numbers contained in the form \(mxx + nyy\)) (E744).  His primary approach in this paper was to note that, for a number of the form \(mx^2+ny^2,\) where \(x\) and \(y\) are relatively prime, its prime divisors fall into certain congruence classes mod \(4mn,\) depending on whether (in our notation) \(mn \equiv\) 0, 1, 2, or 3 (mod 4).  Euler stated but did not prove several assertions.  In the following, \(k\) is an odd number relatively prime to \(mn.\)

Note throughout how similar many of these statements are to those included in Euler’s letter to Goldbach, written 36 years earlier.

• If some prime number congruent to \(k\) (mod \(4mn\)) divides some number of the form \(mx^2+ny^2,\) then all primes congruent to \(k\) (mod \(4mn\)) will  divide some number of the form \(mx^2 + ny^2,\) whereas no prime congruent to \(-k\) (mod \(4mn\)) will divide such a number.  Conversely,  if some prime number congruent to \(k\) (mod \(4mn\)) does not divide any number of the form \(mx^2+ny^2,\) then no prime congruent to \(k\) (mod \(4mn\)) will  divide any number of the form \(mx^2 + ny^2,\) whereas all primes congruent to \(-k\) (mod \(4mn\)) will divide some number of the form \(mx^2 + ny^2.\)
• If  \(mn \equiv\) 1 or 2 (mod 4), and some prime number congruent to \(k\) (mod \(4mn\)) divides some number of the form \(mx^2+ny^2\) so that all primes congruent to \(k\) (mod \(4mn\)) will divide some number of the form \(mx^2 + ny^2,\) then all primes congruent to \(2mn-k\) (mod \(4mn\)) will divide some number of the form \(mx^2 + ny^2,\) whereas no prime congruent to \(2mn+k\)  (mod \(4mn\)) will divide such a number.
• If  \(mn \equiv\) 0 or 3 (mod 4), and some prime number congruent to \(k\) (mod \(4mn\)) divides some number of the form \(mx^2+ny^2\) so that all primes congruent to \(k\) (mod \(4mn\)) will divide some number of the form \(mx^2 + ny^2,\) then all primes congruent to \(2mn+k\) (mod \(4mn\)) will divide some number of the form \(mx^2 + ny^2,\) whereas no prime congruent to \(2mn-k\)  (mod \(4mn\)) will divide such a number.
• All primes congruent to \(k^2\) (mod \(4mn\)) will divide some number of the form \(mx^2 + ny^2,\) whereas no prime congruent to \(-k^2\) (mod \(4mn)\) will divide such a number.

We next see the reappearance of facts reminiscent of the set \(S\) described in our section (page), Backgound: Some Eulerian History, of this paper:

• Let \(p\) be any prime number less than \(mn\) and relatively prime to \(mn.\)  The prime \(p\) will divide some number of the form \(mx^2 + ny^2\) if and only if \(p\) divides \(mn+y^2,\) where \(y\) is a positive integer which is less than or equal to  \(\frac{1}{2} mn.\)  Thus, the primes \(p\)  not dividing such a number \(mn+y^2\)  do not divide numbers of the form \(mx^2 + ny^2.\)  We can, therefore, create a list of the congruence classes mod \(4mn\) which contain primes that divide \(mx^2 + ny^2\) in this way: list the integers \(mn+y^2,\) where \(1\leq y \leq \frac{1}{2} mn;\) find their odd prime divisors \(p\) which are less than and relatively prime to \(mn;\)  then \(p\) (mod \(4mn\)) will represent one of these congruence classes.  Now suppose \(p\) is one of the other odd primes, those which are less than and relatively prime to \(mn\)  but do not divide any of the integers \(mn+y^2.\)  Then \(p\) (mod \(4mn\)) will not be one of these congruence classes, but \(-p\) (mod \(4mn\)) will be one of these congruence classes, for reasons stated above.
• As for the classes  congruent to \(k\) (mod \(4mn\)) where \(k\) is composite, we can determine whether or not these contain primes that divide \(mx^2 + ny^2\) by using the fact that this set is closed under multiplication modulo \(4mn.\)  For example, if the classes congruent to \(-p_1\) and  \(-p_2\) contain such primes, where \(-p_1\) and  \(-p_2\) are not necessarily distinct, then the class \((-p_1)(-p_2)=p_1 p_2 \) also contains such primes.

Finally, Euler stated an assertion which didn't seem to arise naturally as a generalization of his earlier work:

• If \(mn \equiv\) 1 or 2 (mod 4), and integers in the conguence class \(k\) (mod \(4mn\)) divide numbers of the form \(mx^2+ny^2,\) then all integers congruent to \(2mn-k\) (mod \(4mn\)) will also divide numbers of the form \(mx^2 + ny^2,\) whereas if \(mn \equiv\) 0 or 3 (mod 4), and integers in the conguence class \(k\) (mod \(4mn\)) divide numbers of the form \(mx^2+ny^2,\) then all integers congruent to \(2mn+k\) (mod \(4mn\)) will also divide numbers of the form \(mx^2 + ny^2.\)

Read the authors' translation of Euler's E744.

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®.

References

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