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.).

Questions about perfect powers, which are defined by:

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, perfect cube, etc.).

A related theorem is Catalan's Conjecture (now proved), stating that (given integers $x,y,a,b>1$)

$$x^y-a^b=1\iff (x,y,a,b)=(3,2,2,3)$$

Pillai's Conjecture is a conjecture that concerns whether every difference (not only $1$) of perfect powers occurs only finitely often.

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Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$

About a month ago, I got the following : For every positive rational number $r$, there exists a set of four positive integers $(a,b,c,d)$ such that $$r=\frac{a^\color{red}{3}+b^\color{red}{3}}{c^\color{red}{3}+d^\color{red}{3}}.$$ For $r=p/q$…
mathlove
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Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

I've known that one can arrange all the numbers from $1$ to $\color{red}{15}$ in a row such that the sum of every two adjacent numbers is a perfect square. $$8,1,15,10,6,3,13,12,4,5,11,14,2,7,9$$ Also, a few days ago, a friend of mine taught me that…
mathlove
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Is this a new method for finding powers?

Playing with a pencil and paper notebook I noticed the following: $x=1$ $x^3=1$ $x=2$ $x^3=8$ $x=3$ $x^3=27$ $x=4$ $x^3=64$ $64-27 = 37$ $27-8 = 19$ $8-1 = 7$ $19-7=12$ $37-19=18$ $18-12=6$ I noticed a pattern for first 1..10 (in the above…
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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 hopefully for a more elementary method. I can…
Calvin Lin
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What is the smallest integer greater than 1 such that $\frac12$ of it is a perfect square and $\frac15$ of it is a perfect fifth power?

What is the smallest integer greater than 1 such that $\frac12$ of it is a perfect square and $\frac15$ of it is a perfect fifth power? I have tried multiplying every perfect square (up to 400 by two and checking if it is a perfect 5th power, but…
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Integers $n$ for which the digit sum of $n$ exceeds the digit sum of $n^5$

This question is strongly inspired by The smallest integer whose digit sum is larger than that of its cube? by Bernardo Recamán Santos. The number $n=124499$ has digit sum $1+2+4+4+9+9=29$ while its fourth power $n^4=240250031031001002001$ has a…
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$\lfloor x^n\rfloor\lfloor y^n\rfloor$ is a perfect square

Let $x,y\ge 1$ be non-integer real numbers such that $\lfloor x^n\rfloor\lfloor y^n\rfloor$ is a perfect square for all natural numbers $n$. Does it follow that $x=y$? From this question we know the condition under which $\lfloor a\rfloor\lfloor…
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$\gcd(b^x - 1, b^y - 1, b^ z- 1,\dots) = b^{\gcd(x, y, z,\dots)} -1$

Possible Duplicate: Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$ $b$, $x$, $y$, $z$, $\ldots$ are integers greater than 1. How can we prove that $$ \gcd (b ^ x - 1, b ^ y - 1, b ^ z - 1 ,\dots)= b ^ {\gcd (x, y, z, \dots)} - 1\ ? $$
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To find all $n$ such that $(n-1)!+1$ is a perfect power of $n$

How to find all positive integers $n$ such that $(n-1)!+1$ can be written as $n^k$ for some positive integer $k$?
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A $\frac{1}{3}$ Conjecture?

Question: Let $A(n)$ be a finite square $n \times n$ matrix with entries $a_{ij}=1$ if $i+j$ is a perfect power; otherwise equals to $0$. Is it true that $${1 \above 1.5 pt n^2}\sum_{i=1}^n \sum_{j=1}^n a_{ij} \leq {1 \above 1.5pt 3}$$ with…
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Solutions to $A^N+B^N=C^N \pm 1$

Is there a solution to $A^N+B^N=C^N \pm 1$ where $A,B,C,N\in\Bbb{N}$, such that $A,B,C > 1,N \geq 4$ and $gcd(A,B)=gcd(B,C)=gcd(A,C)=1$? This question was inspired by this one: $A^X+B^Y=C^Z\pm 1$ Beal's conjecture "almost" solutions A related…
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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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Perfect powers of successive naturals: Can you always reach a constant difference?

I was thinking about what happens if you take a sequence of consecutive squares, for example 1,4,9, 16. Taking the differences gives you another sequence, 7,5,3. And taking the differences between those numbers, you get 2,2--a constant. Through…
Dark Penguin
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Find all natural number $n$ for which $3^9+3^{12}+3^{15}+3^n$ is a perfect cube.

Find all natural number $n$ for which $3^9+3^{12}+3^{15}+3^n$ is a perfect cube. What I have tried. $3^9+3^{12}+3^{15}+3^n$ $=3^9(757+3^{n-9})$ Let $757+3^{n-9}=a^3$ Taking modulo $3$: $a^3\equiv 1 \pmod 3$ $\implies a\equiv 1 \pmod 3$ Also,…
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Can a prime have arbitary many representations as a sum of two perfect powers?

Let $p$ be a prime number and $f(p)$ be the number of representations $$p=a+b$$ with perfect powers $0
Peter
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