Euler's factorization method

Euler's factorization method is a technique for factoring a number by writing it as a sum of two squares in two different ways. For example the number ${\displaystyle 1000009}$ can be written as ${\displaystyle 1000^{2}+3^{2}}$ or as ${\displaystyle 972^{2}+235^{2}}$ and Euler's method gives the factorization ${\displaystyle 1000009=293\cdot 3413}$.

The idea that two distinct representations of an odd positive integer may lead to a factorization was apparently first proposed by Marin Mersenne. However, it was not put to use extensively until one hundred years later by Euler. His most celebrated use of the method that now bears his name was to factor the number ${\displaystyle 1000009}$, which apparently was previously thought to be prime even though it is not a pseudoprime by any major primality test.

Euler's factorization method is more effective than Fermat's for integers whose factors are not close together and potentially much more efficient than trial division if one can find representations of numbers as sums of two squares reasonably easily. Euler's development ultimately permitted much more efficient factoring of numbers and, by the 1910s, the development of large factor tables going up to about ten million. The methods used to find representations of numbers as sums of two squares are essentially the same as with finding differences of squares in Fermat's factorization method.