The Search for One as a Prime Number: from Ancient Greece to Modern Times

It has often been asked if one is a prime number, or if there was a time when most mathematicians thought one was prime. Whether or not the number one is prime is simply a matter of definition, but definitions are often decided by the use of mathematics. In this paper we will survey the history of the definition of prime as applied to the number one, from the ancient Greeks to the modern times. For the Greeks the numbers (αριθμoς) were multiples of the unit, and for this reason one did not fall into the category of primes (a subdivision of the numbers). This view held with few exceptions until Stevin (c. 1585) argued successfully that one was a number, at which point it finally made sense to ask if one is prime. This was followed by a period of confusion which began to dissipate with Gauss’ Disquisitiones Arithmeticae. Our survey will show that for most of history, one was not considered a prime, and there was no point in time where a clear majority of mathematicians viewed one as prime.