Math is funny. There's an infinite amount of primes, and even though we know a lot of them there's still more left (read:infinite). But at the same time, with high-school level algebra we can prove that the largest prime followed by a cubic number is 7.<p>The proof is really simple, but the fact is quite unintuitive (at least for me).
Huh, I never realized this was this simple. Fleshing out the general case the post mentions.<p>Suppose p=x^n-1 is prime.<p>Note that:<p><pre><code> x^n - 1 = (x^n + x^(n-1) +... + x^2 + x)
-( x^(n-1) +... + x^2 + x +1)
=>x^n-1 = (x^(n-1) + x^(n-2) + ... + x + 1)(x-1)
</code></pre>
So if p=x^n-1 is prime, one of the factors must be equal to 1. If the first factor is 1, this implies x=0 and so the second factor is -1. Therefore the second factor must be 1, and x=2.<p>Therefore if x^n-1 is prime, x must be 2. Damn that's cool. At first I thought any number 2^n-1 would be prime, but then I realized that obviously the left hand factor can have subfactors. Which is to say I remembered 15 exists.
The proof states: "For n to be prime either x² + x + 1 = 1 or x - 1 = 1"<p>Could somebody tell me why each of the factor components must be equal to one?