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. 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.
As he admitted to Goldbach in , Euler couldn't prove any of these statements at the time. He would prove Theorem 3. Euler's inability to prove these statements, however, did not stop him from stating a more general version of these ideas.
The resulting statement is crucial to our present discussion, and we shall restate it more formally:. Claim 3. 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?
Not only the language and notation, but even the standards of proof and description have changed dramatically over the centuries. Euler's Claim 3. 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!
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. 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. 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. Edwards, in his article, included a discussion about how one would prove a more standard version of quadratic reciprocity from Euler's Claim 3. 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. Theorem 5. 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.
In summary, to go from his published work to quadratic reciprocity, Euler would need Theorems 5. 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.
It contains an exciting new generalization of the types of quadratic forms Euler had studied in E 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 E in , and it was not published until —a 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 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 E 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 E in , kicking off a three-year period in which he returned with gusto to the study of Diophantine equations. 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.
Yet in some sense, E is disappointing. By almost any measure, Euler was by the most famous and accomplished mathematician in Europe. One of his many! Like many of his lifelong projects cf. 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. 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, E is a successful simplification and generalization of the work he did earlier in such papers as E [4], E [5], and E [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 E 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. Euler stated but did not prove several assertions. Finally, Euler stated an assertion which didn't seem to arise naturally as a generalization of his earlier work:.
A statement of Lemma 5. This is such a great book it was the course text for a course on Cryptography I took last spring and I'd recommend it to anyone who is interested in learning more about the practical and totally AWESOME applications of Algebraic Number Theory.
There is so much more to be said about Quadratic Reciprocity mentioned in the end of the 3rd chapter in this book , so be sure to check it out! There is a very nice historical introduction at wikipedia. This shows how mathematicians started to think in the direction of quadratic reciprocity after seeing the work of Fermat, and later how Legendre "almost" proved it, etc. A lot of credit has to be attributed to Legendre. It is not Gauss alone who made contributions to arithmetic during that period.
In fact, in the preface to his "Disquisitiones", Gauss acknowledges the work of Legendre. Sign up to join this community.
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Asked 11 years ago. Active 2 years, 4 months ago. Viewed 11k times. So far I can think of two uses that are basic enough to be shown immediately when presenting the theorem: 1 With the QRT, it is immediate to give a simple, efficient algorithm that can be done even by hand for computing Legendre symbols. Gadi A Gadi A If that is correct the title should be changed. Add a comment. Active Oldest Votes. Pete L.
Clark Pete L. Clark Qiaochu Yuan Qiaochu Yuan k 41 41 gold badges silver badges bronze badges. Is it known that quadratic reciprocity even special cases was not independently discovered e. For India, I think Andre Weil, in his second career as math historian and Sanskrit scholar, would have told us, e. Maybe the theory around sums of two squares is accessible enough to be universal, but beyond that? Tanner 1. Mell Miller Mell Miller 39 2 2 bronze badges. Sign up or log in Sign up using Google.
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