# Gauss's proofs of quadratic reciprocity

Gauss gave 6 published and 2 unpublished proofs of quadratic reciprocity (see, e.g., here). I suspect this was to try to understand the "real reason" quadratic reciprocity holds (though please correct me if you know otherwise), but I'd like to know what Gauss actually thought about his different proofs. Do we know to what extent Gauss was satisfied with his proofs, or which proofs he regarded as better than others?

• I totally agree with you that his motivation was “to try to understand the "real reason" quadratic reciprocity holds”. – José Carlos Santos Oct 20 '17 at 8:16
• Wasn't Gauss kinda famous for never being satisfied with his proofs because they weren't "elegant enough" ? – Carl Witthoft Oct 20 '17 at 12:44
• Where do you get that idea from? I think he's primarily famous for his motto 'pauca sed matura' (lit. 'few but ripe'), and for living up to it. – RP_ Oct 20 '17 at 13:26
• @René I guess "elegant" <--> "few" here. – Carl Witthoft Oct 20 '17 at 18:50

Yes, Gauss wrote in several places about his proofs of the reciprocity theorem. Here's one really interesting passage from 1808, worth quoting at length:

[Arithmetical truths] are frequently of such a nature that they may be arrived at by many distinct paths and that the first paths to be discovered are not always the shortest. It is therefore a great pleasure after one has fruitlessly pondered over a truth and has later been able to prove it in a round-about way to find at last the simplest and most natural way to its proof. ...For a whole year [the reciprocity theorem] tormented me and absorbed my greatest efforts until at last I obtained a proof given in the fourth section of [the Disquisitiones]. Later I ran across three other proofs which were built on entirely different principles. One of these I have already given in the fifth section; the others, which do not compare with it in elegance, I have reserved for future publication. Although these proofs leave nothing to be desired as regards rigor, they are derived from sources much too remote, except perhaps the first, which however proceeds with laborious arguments and is overloaded with extended operations. I do not hesitate to say that till now a natural proof has not been produced. I leave it to the authorities to judge whether the following proof which I have recently been fortunate enough to discover deserves this description. (Translated in David Eugene Smith, A Source Book in Mathematics, Volume One (Dover, 1959), 113.)

When Gauss wrote this, he'd previously published two proofs of quadratic reciprocity (the ones appearing in DA -- the first by induction, and the second via his new theory of quadratic forms). He'd also found two more proofs which wouldn't be published until after his death (involving quadratic periods). Here, he's about to introduce what's usually known as his third proof. This is the one using Gauss's Lemma that Eisenstein would later improve and geometrize, turning it into the "lattice point counting" proof that you often see in textbooks today.

Gauss's third proof was apparently his favorite, as the passage shows. He found the earlier proofs unsatisfactory because they were either too complicated or "derived from sources much too remote". The third proof was the only one he considered natural.

Interestingly, although Gauss is often (and rightly) praised for his deep mathematics, his ideal proof was evidently a simple and transparent one rather than one of great profundity and abstractness.

The other proof about which he had some approving things to say was the "Gauss sums" proof (usually called the sixth proof due to publication order, although it was the eight and last to be discovered). This is the other one that crops up a lot today, in part because it's relatively easy to generalize to get proofs of cubic and quartic reciprocity. Gauss was very interested in finding a proof like this, which is one reason he kept looking for new approaches to QR:

From 1805 onwards I have investigated the theory of cubic and biquadratic residues... Theorems were found by induction... which had a wonderful analogy with the theorems for quadratic residues. On the other hand, for a long time all attempts at complete proofs have been futile. This was the motive for endeavoring to add yet more proofs to those already known for quadratic residues, in the hope that of the many different methods given, one or the other would contribute to the illumination of the related arguments [for higher reciprocity]. (Translated in David Cox, Primes of the Form $$x^2+ny^2$$, 2nd edition (Wiley, 2013), 78).

Here's what Gauss has to say about the sixth proof, which finally gave him what he wanted:

Although the arithmetical theorem dealt with here [i.e., QR] has, by earlier efforts, been provided with four entirely different proofs, and can seem to be completely finished, I return here to some new observations and present two new proofs, which shed a new light on these facts... [T]he sixth proof calls upon a completely different and most subtle principle, and gives a new example of the wonderful connection between arithmetic truths that at first glance seem to lie very far from one another... Still another reason was provided, which led me to publish a new proof now that I had already spoken of 9 years earlier. In fact, as I was occupied in 1805 with the theory of cubic and biquadratic residues, where a very difficult circumstance that I had begun to work through, happened to draw on almost the same skill as originally in the theory of quadratic residues. Without further notice, indeed, were those theorems -- which had handled those questions completely, and had presented a wonderful analogy with the corresponding theorems about quadratic residues -- found by induction as soon as they were only sought for in a proper way, and all remaining researches led on all sides to a complete proof that for a long time I had sought in vain. This was indeed the impulse that I so much sought to add more and more proofs of the already-known theorems on quadratic residues, in the hope that from these many different methods, one or another could illuminate something in the related circumstances. This hope was in no way idle, and untiring efforts were finally crowned with success. Shortly, I will be in a position to publish the fruits of my studies, but before I undertake that difficult work I have decided to return once more to the theory of quadratic residues, and to say what is still to be said, and so in a certain sense to bid farewell to this part of the higher arithmetic. (Translated in Jeremy Gray, A History of Abstract Algebra: From Algebraic Equations to Modern Algebra (Springer, 2018), 333).)

Actually, Gauss never got around to publishing proofs of cubic or quartic reciprocity, and it seems to be unclear to what extent he had them fully worked out. But he at least realized that the sixth proof was the right place to start.

• Thanks very much! Re: Interestingly, although Gauss is often (and rightly) praised for his deep mathematics, his ideal proof was evidently a simple and transparent one rather than one of great profundity and abstractness. I don't think this is anything uncommon or surprising. Indeed, one of the reasons mathematicians pursue abstraction is to make reasons transparent. (I would also argue that profundity is not at odds with simplicity, but won't pontificate on whether the 3rd proof is profound.) – Kimball Jun 24 '19 at 4:51