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

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.