## moneybird Group Title Perfect power question 2 years ago 2 years ago

1. moneybird Group Title

For what primes p is $3^p + 4^p$ a perfect power?

2. asnaseer Group Title

for p=2 we get:$3^2+4^2=5^2$

3. moneybird Group Title

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))

4. Stom Group Title

but can there be other such primes too?

5. moneybird Group Title

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

6. mukushla Group Title

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

7. mukushla Group Title

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)