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Did Euler Know Quadratic Reciprocity?: New Insights from a Forgotten Work - The Debate

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

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.

Paul Bialek (Trinity International University) and Dominic W. Klyve (Central Washington University), "Did Euler Know Quadratic Reciprocity?: New Insights from a Forgotten Work - The Debate," Convergence (February 2014)