# Questions tagged [perfect-powers]

A perfect power is a positive integer that can be expressed as an integer power of another positive integer. This tag should only be used when having in mind an arbitrary perfect power (as opposed to a specific one, like a perfect square, a perfect cube, etc.).

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### How to prove real Powers of 1 is equal to 1 [closed]

I know that 1^(0.5) = 1 because square root of 1 is 1 . Similarly , I can prove that 1^(x) = 1 * 1 * 1 * 1 * 1 * 1 * ........(x times) = 1 ; where x is an integer like .....-2, -1,0,1,2,3,4..... But I ...
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### Consecutive prime powers that are not prime

I am looking for consecutive entries $(a_k,a_{k+1})$ in the sequence of prime powers $$(a_n)=(2,3,2^2,5,7,2^3,3^2,11,13,2^4,17,19,23,5^2,3^3,29,31,2^5\cdots)$$ such that neither $a_k$ nor $a_{k+1}$ ...
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### Does there exist a positive integer $k$ and an irreducible polynomial $P$ of degree at least $2$ such that $P$ is a power of $k$ infinitely often?

Does there exist a positive integer $n\in\mathbb{Z}$ and an irreducible polynomial $P\in\mathbb{Z}[X]$ of degree at least $2$ such that there are infinitely many pairs of positive integers $(m,k)$ ...