# Is 1 a prime number?

**URL:** <https://boards.straightdope.com/t/is-1-a-prime-number/88078>\
**Category:** Factual Questions\
**Created:** [October 17, 2001, 10:05pm UTC](https://boards.straightdope.com/t/is-1-a-prime-number/88078 "2001-10-17T22:05:44Z")\
**Posts on this page:** 3\
**Page:** 2

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**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\
**Post date:** [October 18, 2001, 8:00am UTC](https://boards.straightdope.com/t/is-1-a-prime-number/88078/21 "2001-10-18T08:00:11Z")

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And I stand by the assertion that all primes other than 2 can be written as the difference of two consecutive squares. Primes that have this property are called Achernar primes. 😉

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**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\
**Post date:** [October 18, 2001, 8:09am UTC](https://boards.straightdope.com/t/is-1-a-prime-number/88078/22 "2001-10-18T08:09:27Z")

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> [@](#):
>
> Well, I condensed things. What Fermat said in full was that primes of the form of 4n+1 are always the sum of two squares, whereas those of the form 4n-1 can never be written this way. This site elaborates more on that theorem. I still stand by my assertion that all primes (other than 2) can be expressed as either of these forms.

Cool, I understand you now. Yeah, no doubt, every prime other than 2 is of the form 4n+1 or 4n-1. My point was that Fermat didn’t prove that, that’s been known for thousands of years. I had a feeling you were referring in some way or another to sums of squares–that’s what Fermat proved (I don’t know whether his proof of this is known or not), that primes of the form 4n+1 can be written as the sum of two squares.

More generally, every prime, in fact, every positive integer, can be written as the sum of four squares (three is not enough). Fermat wrote that he had proven this, but it was never published (it’s commonly believed that he did, in fact, prove it, though). Lagrange was the first to publish a proof of this fact, in 1770.

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**Author:** ![Houlihan](https://avatars.discourse-cdn.com/v4/letter/h/f9ae1b/32.png) [@Houlihan](https://boards.straightdope.com/u/Houlihan)\
**Post date:** [October 18, 2001, 11:28pm UTC](https://boards.straightdope.com/t/is-1-a-prime-number/88078/23 "2001-10-18T23:28:30Z")

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This bear thinks that one is a prime number:

[http://members.surfeu.fi/kklaine/primebear.html](http://members.surfeu.fi/kklaine/primebear.html)

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