E708 -- De formulis speciei mxx + nyy ad numeros primos explorandos idoneis earumque mirabilibus proprietatibus
(On forms of the type mxx + nyy, for exploring prime numbers by idoneals of them with remarkable properties)
"If m and n are positive numbers and if N is contained in two different manners in the form mxx + nyy, so that N = maa + nbb and N = mcc + ndd, in reducing the fraction (a ± c)/(b ± d) to its lowest terms, then the fraction (mpp)/(nqq) = r/s; in the case r + s is an odd number, it will be a factor of the number N; if it is even, then its half, (r + s)/2 will be a factor of that number N. From this, it follows that any number contained in more than one fashion in the form mxx + nyy will be a composite, or non-prime number." Euler goes on to define "congruent" numbers, describe their properties, and then give the famous list of the 65 congruent numbers, all less than 10000. Feuter warns that the proofs of theorems 6, 8, 9 are "not rigorous."
Originally published in Nova Acta Academiae Scientarum Imperialis Petropolitinae 12, 1801, pp. 22-46
Opera Omnia: Series 1, Volume 4, pp. 269 - 289
- Reprinted in Commentat. arithm. 2, 1849, pp. 249-260 [E708b]
- Original Publication: E708
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