E715 -- De variis modis numeros praegrandes examinandi, utrum sint primi necne?

(On various ways of examining very large numbers, for whether or not they are primes)


Euler reviews the fact that if a number has two representations as axx + byy, then it cannot be prime, 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.

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