Consider what happens if n is not a prime number, so n = pq. You can show that is divisible by . Specifically:
Therefore cannot be prime unless , and that can only happen if a = 4 and p=1.
How could you prove that if a^n-3^n is a positive prime number, then a=4 and n is a positive prime number? (all numbers are natural)
Am I supposed to try and factor a^n-b^n to see for which a and b the factorization is possible and for which it's not? I am having trouble with that factorization anyhow, so does anyone know how to approach this problem?
I mean it seems to work for small n, but I just can not figure out how to do it for all natural numbers n...
Thankful for answer!
Consider what happens if n is not a prime number, so n = pq. You can show that is divisible by . Specifically:
Therefore cannot be prime unless , and that can only happen if a = 4 and p=1.
Sorry I was being unclear. I understand that it is true, but is it enough to say that a^{pq} - b^{pq} = (a^p)^q-(b^p)^q to clarify that (a^p-b^p) is in fact a factor? Since I'm unsure I think it's not ^^. Could you maybe explain a bit further in what way the rewriting (a^p)^q-(b^p)^q leads to the answer? Unless it's not done by that rewriting ^^
/Pkaff