Lol check this out https://www.google.com/search?q=is+-1+prime

More seriously

http://mathforum.org/library/drmath/view/55958.html

But the link to details is dead.

Internet Archive suggests the original discussion was a Usenet thread; here is the link to Google Groups' version of it:

https://groups.google.com/d/topic/geometry.research/7pyFhAAy...

> Lol check this out https://www.google.com/search?q=is+-1+prime

I saw that, looking for references... very amusing. Even funnier (to me):

https://www.google.com/search?q=is+1000000101110000000000000...

-1 can't be prime because many things that use prime numbers require a positive number. For example, how would you define prime powers? (-1)^2=1...

You can certainly give a name to the integers {-1} U P, but maybe it would be better to call them "choice" or "select" numbers.

As John Conway said (see sibling comment, or e.g., [1]):

> Every nonzero rational number has a unique factorization into powers of distinct primes.

As you note, (-1)^2 = 1. But if you read carefully, you'll see that the factorization of -100/3 is uniquely:

  -1 * 4 * 1/3 * 5 * 1 ...

whereas the factorization of 100/3 is uniquely:

  1 * 4 * 1/3 * 5 * 1 ...

Where it's true that the representation using primes with exponents is not unique, it is true that the representation using powers of primes is unique. That is, your issue regarding (-1)^2 is that there are infinitely many representations of 1 or -1 having the form (-1)^x, but if you evaluate (-1)^x (x being integral, of course) you'll only get one of two numbers.

And yes, changing the definition of "prime" does change some special cases -- it removes some (such as extending unique factorization to negative rationals), and adds others (wherever primes are assumed positive).

[1] http://swc-alpha.math.arizona.edu/video/2009/2009ConwayLectu... (mention around 7:00)

Mr Conway was (RIP) a genius. If he says -1 is prime ... it's prime. End of.