# 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.).

191 questions
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### Integer solutions to $2^x-1=3^y$.

Does anybody know how to solve it? I know how to solve linear diophantine equations, but equation like this I've never seen before. $$2^x-1=3^y.$$
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### Perfect powers of the form $2^n\pm n$?

In the range $$1\le n\le 10^5$$ the only perfect powers of the form $$2^n\pm n$$ are $$2^5-5=3^3$$ and $$2^7-7=11^2$$ How can I prove that there are no more perfect powers of this form ? The case ...
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### $A^X+B^Y=C^Z\pm 1$ Beal's conjecture “almost” solutions

Beal's conjecture is a generalization of fermat's last theorem. Fermat's last theorem states that there are no solutions to the equation $A^N+B^N=C^N$ where $A,B,C,N\in \Bbb{N}\space |\space N\ge 3$ ...
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### Can any Power function (whose base is not zero) , e.g. $2^{n}$, be defined arithmetically (i.e. using addition and multiplication only)? [duplicate]

In other words, I'm looking for a binary relation $P(x,y)$, being arithmetical, i.e being expressed in the first order language of Peano arithmetic (hence non-recursively, i.e. using addition and ...
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