One is the Loneliest Number
from the song One, by Three Dog Night

Some people believe that one should also be a prime number. A very few teachers even teach their students that one is a prime number. They think that the definition applies to 1. After all, 1 is divisible by itself and it is also divisible by 1. Don't believe either of them. 1 is not a prime number because it does not have exactly two DIFFERENT factors.

If you have read the definitions page, then you have seen that every reference book and every math textbook that I checked agrees. One is not a prime number.

In some cases, the definition specifically states that 1 is not a prime number. In other cases, the definition requires exactly two factors. Some definitions don't explicitly require two factors, but tell me, if a friend told you

"She has only Jane and herself to blame"

how many people would she have to blame? Right, two. So, a definition of "only factors are itself and 1" certainly implies that 1 is not a prime number. It would say "only factors are itself and/or 1".

Let's look at Euclid's definitions again (from Euclid's Elements: Book VII and Book IX). For convenience, I will rephrase what he said into a more normal English.

• A unit of any thing that exists is called one
• A number is a multitude composed of units (by this statement, Euclid states that one is not a number, something that shows up in some of his later proofs.
• A greater number is a multiple of a lesser number when it can be measured by the lesser number
• A prime number is that which is measured by an unit alone (combined with the second bullet, we see that 1 cannot be measured by units. It is excluded above).
• "a number measuring the unit is absurd" (Euclid's second statement that 1 is not a number).
• One is "unity" and not a number by Euclid's mathematics.

All definitions agree. 1 is not a prime number.

Here's some information from AACDrPhil that appears to support Euclid's statement that 1 is not a number, and thus not a prime number:

However, when reading some of the proofs of prime number theorems, especially the famous Book IX, Proposition 20 (that there are always more primes), I see that "... a number measuring the unit is absurd." It looks like Euclid's definition of "number" excludes 1 -- a number is a "multitude" of units. So if 1 is not a number, it is also not a prime number. If Euclid had considered 1 to be a prime number, his proof of IX 20 would have to account for it explicitly (which it easily could have). It also appears that Euclid did not consider a number to be measured by itself.

(AACDrPhil, PhD, Astro-Nuclear-Bio-Physics volunteer teacher)

To be fair to Phil, I should point out that he and I have been corresponding on this subject, and he feels that one could be included as a prime number, even though by current definitions it is not. In some cases on this page, I am using his arguments for one being included as a prime number as part of my argument against the same. Thanks, Phil, for the fascinating correspondence. And yes, we are having fun with our back-and-forth e-mail.

You can also see that one is not a prime number when you use the "Sieve of Eratosthenes", where you circle each prime number, then cross off every multiple of that prime number.

• If you were to circle ONE as a prime number, and then cross off every multiple of ONE, you would cross off every other number.
• The "Sieve of Eratosthenes" will not work if one is a prime number.

I hope that all of this information, the definitions and this math alert, helps everyone to realize that ONE is not a prime number.

I understand that a hundred years or so ago, some books actually said that one was a prime number. I've never seen such a book, but I'd love to see it. If you have such a book, feel free to send me a copy of what you have. My address is below. Be sure to also send me a copy of the title page and the copyright page along with the copy of the page with the details. Thanks.

If you are a publisher and have a book that you would like included on our definitions page (and on other pages I eventually write), you may send the book to: