An inquiry into whether or not 1000009 is a prime number

Abstract

Translation from the Latin original "Utrum hic numerus 1000009 sit primus necne inquiritur" (1778). E699 in the Enestrom index. The idea of this paper is that if some number is a sum of two squares in two ways, then some other smaller number must be a square. We then check all the cases. If we find a case where this smaller number is a square then we can use this to find a factor of the original number, which is therefore composite. If we check all the cases and none of them are squares then the original number can be written as a sum of two squares in only one way, and hence is prime. Euler first examines the number 1000009 and then 1000081. Obviously 1000009=1000^2+3^2, and then he finds that 1000009=972^2+235^2, hence it is not prime. On the other hand, Euler finds that 1000081 can be written as a sum of squares in only one way, hence it is prime.