On various ways of examining very large numbers, for whether or not they are primes
Euler reviews the fact that a number with two representations of the form axx+byy cannot be prime, then demonstrates the theorem by applying it to 12091 = axx+byy for (x, y) = (40, 9) and (4, 33), and using this to factor 12091. He proves that if M and N are of the same form, then so is MN. He presents the problem: given a formula axx+byy and a number that can be written in that form in only one way, when can you conclude that the number must be prime? This reduces to a problem in congruent (idoneal) numbers, and Euler gets to repeat his table of 65 such numbers.
Original Source Citation
Nova Acta Academiae Scientiarum Imperialis Petropolitanae, Volume 13, pp. 14-44.
Opera Omnia Citation
Series 1, Volume 4, pp.303-328.