Question

Perfect power question

Answer 1

For what primes p is
\[3^p + 4^p\] a perfect power?

Answer 2

for p=2 we get:\[3^2+4^2=5^2\]

Answer 3

yeah 2 is the only even prime too.

so you can factor 3^p + 4^p to (3+4) (3^(p-1) - 4 * 3^(p-2) ... + 4^(p-1))

Answer 4

but can there be other such primes too?

Answer 5

Other than p = 2, 3^p + 4^p must be divisible by 7 i think

Answer 6

so we lookin for the solutions of this equation\[3^p+4^p=q^n\]p is prime and q,n>1

Answer 7

for p>7
\[3^p+4^p\]is divisible by \(7\) but not by \(7^2\) since\[3^{p-1}-4\times3^{p-2}+4^2\times3^{p-3}-...-3\times4^{p-2}+4^{p-1} \ \ \equiv p3^{p-1} \ \ \text{mod} \ 7\]so there is no solution for p>7

checking for p=2,3,7 gives only solution (p,q,n)=(2,5,2)