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anonymous
 4 years ago
Perfect power question
anonymous
 4 years ago
Perfect power question

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anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0For what primes p is \[3^p + 4^p\] a perfect power?

asnaseer
 4 years ago
Best ResponseYou've already chosen the best response.0for p=2 we get:\[3^2+4^2=5^2\]

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0yeah 2 is the only even prime too. so you can factor 3^p + 4^p to (3+4) (3^(p1)  4 * 3^(p2) ... + 4^(p1))

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0but can there be other such primes too?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Other than p = 2, 3^p + 4^p must be divisible by 7 i think

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0so we lookin for the solutions of this equation\[3^p+4^p=q^n\]p is prime and q,n>1

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0for p>7 \[3^p+4^p\]is divisible by \(7\) but not by \(7^2\) since\[3^{p1}4\times3^{p2}+4^2\times3^{p3}...3\times4^{p2}+4^{p1} \ \ \equiv p3^{p1} \ \ \text{mod} \ 7\]so there is no solution for p>7 checking for p=2,3,7 gives only solution (p,q,n)=(2,5,2)
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