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Did Euler Know Quadratic Reciprocity?: New Insights from a Forgotten Work - The Contents of 'De divisiborus numerorum'

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

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.

Paul Bialek (Trinity International University) and Dominic W. Klyve (Central Washington University), "Did Euler Know Quadratic Reciprocity?: New Insights from a Forgotten Work - The Contents of 'De divisiborus numerorum'," Loci (February 2014)

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