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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Product of three consecutive positive integers is never a perfect power

I am trying to prove that the product of three consecutive positive integers is never a perfect power. Can anyone point to gaps in my proof and/or post an alternate solution? Let the three positive ...
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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 ...
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Who can prove that a triangular number cannot be a cube, fourth power or fifth power?

Triangular numbers (See https://en.wikipedia.org/wiki/Triangular_number ) are numbers of the form $$\frac{n(n+1)}{2}$$ In ProofWiki I found three claims about triangular numbers. The three claims ...
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When is $2^n \pm 1$ a perfect power

Is there an easy way of showing that $2^n \pm 1$ is never a perfect power, except for $2^3 + 1 = 3^2$? I know that Catalan's conjecture (or Mihăilescu's theorem) gives the result directly, but I'm ...
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$x^2+x+1$ is the cube of a prime.

Please help me find all natural numbers $x$ so that $x^2+x+1$ is the cube of a prime number.(Used in here)
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If $a + b + c \mid a^2 + b^2 + c^2$ then $a + b + c \mid a^n + b^n + c^n$ for infinitely many $n$

Let $a,b,c$ positive integer such that $a + b + c \mid a^2 + b^2 + c^2$. Show that $a + b + c \mid a^n + b^n + c^n$ for infinitely many positive integer $n$. (problem composed by Laurentiu ...
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Diophantine equation $15^x+8^y=17^z$

How to solve the following equation over the integers? $$15^x+8^y=17^z$$ I know that only solution is $(x,y,z) = (2,2,2)$, but how to prove this?
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$n$-th term of the series 1 27 125 1000

What will be the nth term of the series 1 27 125 1000 for $n = 1$, it is 1 for $n = 2$, it is 27 for $n = 3$, it is 125 for $n = 4$, it is 1000
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Equivalence between a limit and Pillai's Conjecture

I have rewritten this problem entirely please see the edits: On the MSE I asked the following conjecture: * The ${1\above 1.5 pt 3}$ Conjecture:* Let $A(n)$ be a finite square $n \times n$ matrix ...
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Is it possible that each finite sum is a perfect power?

Rough question: Does there exist an infinite set of positive integers such that the sum of any finite subset is a perfect power?
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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 ...
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Are there arbitary large non-trivial powers NOT containing all digits? [duplicate]

Are there arbitary larger numbers of the form $a^b$ with positive integers $a,b>1$ and $a\ne 0\mod 10$ NOT containing all digits ? Here : Biggest powers NOT containing all digits. is either a ...
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$2^W = (2a_1+1)^2 \cdot 2^{M_1}-a_1$ - is there a nontrivial diophantine solution in $M_1,a_1$ given possible?

Update/error: I had a small error in the $(2a+1)$-term in the formula which should have been $(a+1)$ . Because I added now my derivations in long and broad at the end of the question I also rewrite a ...
The answers in Given an integer, how can I detect the nearest integer perfect power efficiently? show two effective algorithms for computing the nearest perfect power for a given integer $n$, although ...
Determine if a positive integer $x$ is a product of a power of 2 and a power of 5. $f(x) = 2^n \cdot 5^n$
Determine if a positive integer $x$ is a product of a power of $2$ and a power of $5$. $f(x) = 2^m \cdot 5^n$ where $0 < x < 32$ and $0 < m < 32$ and $0 < n < 32$ This has to do ...