A Prime Testing Algorithm from Leonhard Euler

Abstract

In 1749, Leonhard Euler solved a longstanding open problem in number theory, proving Fermat’s 1640 conjecture that any prime number of the form can be written as the sum of two squares. This achievement overshadowed an equally impressive calculation that Euler devised at the same time, to test numbers for primality. In this article, we explore Euler’s primality test, demonstrating its computational parity with trial division, at least for a class of testable numbers. Additionally, we will see that primality testing was one of Euler’s lifelong interests, and was a topic he returned to time and again as an application of his number theory work.