(February 28, 2018 at 1:21 pm)robvalue Wrote: Prime numbers are natural numbers p so that:
p>1; If a natural number n>0 divides exactly into p, then n=1 or n=p
Another way of saying this is that each p has exactly two factors among the natural numbers.
So why isn't 1 allowed to be a prime number? It's because we then wouldn't have unique factorization into primes for natural numbers. For example:
12 = 2 * 2 * 3 = 2^2 * 3
is a unique factorization. But if we allow 1 to be prime, we have
12 = 1 * 2^2 * 3 = 1^2 * 2^2 * 3 = 1^3 * 2^2 * 3 =...
And if we go to integers, we have to deal with things like
12=(-2)*2*(-3)
That makes life a bit more interesting!
Past this, there is the notion of a Gaussian integer: a complex number of the form m+n*i where i*i=-1.
Many of the results that are true for integers are also true for Gaussian integers, but which numbers are prime can be different.
For example, 5=(2+i)*(2-i) is no longer prime! But, it turns out, 7 is still prime!
