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Did Euler Know Quadratic Reciprocity?: New Insights from a Forgotten Work - Background: Some Eulerian History

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

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?

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

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