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
2k views

592 views

758 views

How to compute 2-adic square roots?

I know that, for a $2$-adic unit to be a perfect square, it must be of the form $\cdots001.$, for example the number $17$ ($10001.$) is a $2$-adic square. How would I go about finding the $2$ adic ...
185 views

126 views

Is there another prime $p$ such that $S(p)$ is prime?

Denote $$S(p):=2^2+3^3+5^5+\cdots +p^p$$ $S(p)$ is prime for $p=3,7,89$. Is there another prime $p$ such that $S(p)$ is prime ? Is the number of primes $p$ such that $S(p)$ is prime, finite ?
241 views

Prime Powers and Differences of Consecutive Cubes

I am wondering if it has been proven that there does not exist a prime $p$ and an integer $r \ge 3$ such that $p^r = (n + 1)^3 - n^3$ for some integer $n$. Note that this is a special case of Beal's ...
214 views

Prove that $\sqrt {2^n-1}$ is irrational for every integer $n>1$

Prove that $\sqrt {2^n-1}$ is irrational for every integer $n>1$ I tried assuming it was equal to $\frac p q$. I get $2^nq^2-q^2 = p^2$ But I don't see where to go from there.
$s(n^x)$ is not a perfect square for all $x$
Does there exist an $n$ such that $s(n^x)$ is not a perfect square for all positive integers $x$ where $s(m)$ denotes the sum of the digits of a positive integer $m$? If $n = 5$, for example, then ...