#### English Title

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

#### Enestrom Number

715

#### Fuss Index

20

#### Original Language

Latin

#### Published as

Journal article

#### Published Date

1802

#### Written Date

1778

#### Content Summary

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.

#### Record Created

2018-09-25