‘1’ is the Loneliest Number
This week, we suggested a list of prime numbers- included was the number ‘1.’ In quick order there followed:
From Joe Prochaska of Nashville, TN:
One is not a prime number! But please, mention this only softly on the air, as this is one of my favorite bar bets.
From Peter Sirokman of Waltham, MA
There are many prime numbers, but the number 1 has never been among them.
And then this from ‘Audrey’ in East Providence, RI:
You’ll probably receive a ton of mail about this, but in tonight’s show, there was a mental math category in which you listed off the prime numbers. However, you made a mistake in this because you started with the number one. One is not a prime number. A prime number is defined as having exactly two whole number factors: one and itself. One has only one whole number factor: one. It should also be noted that one is not composite, either, because composite numbers have more than two factors.
Thanks to these folks- and the legions who followed- for checking our arithmetic.
Very good. But Mr Sirokman exaggerates slightly. Many mathematicians included 1 in the list of prime numbers through the nineteenth century. In the effort to clean up rough edges of mathematical consistency that began in the nineteenth and proceeded into the twentieth century, those who excluded 1 from the primes prevailed. In particular, the assertion known as the Fundamental Theorem of Arithmetic requires that 1 NOT be a prime number:
FToA: “Every integer greater than 1 is either a prime number or is the product of a unique set of prime numbers.”
If 1 were prime, then for example
15 = 3 x 5 [the only unordered set of prime numbers whose product is 15 is {3, 5} if 1 is NOT prime]
*and also*
15 = 1 x 3 x 5 [and also (1 x 1 x 3 x 5) and (1 x 1 x 1 x 3 x 5) etc. if 1 IS prime],
so the FToA would not hold.