4 ms·

-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 cer

• klyrs 6y ago
  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/2009/2009ConwayLecture1.mp4 http://swc-alpha.math.arizona.edu/video/2009/2009ConwayLectu... (mention around 7:00)