# 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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### Given $a,b$ find perfect cube solutions to $a+bx, \forall x\in\mathbb{N}$ [closed]

Suppose we are given $a,b\in\mathbb{N}$ find $x\in\mathbb{N}: \exists k\in\mathbb{N}: a+bx=k^3$.
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### Number Theory Problem: Find a perfect power k^m > 1 where k, m, k^m do not contain 2 in their decimal digits, nor do share any decimal digit.

Find a perfect power k^m > 1 where k, m, k^m do not contain 2 in their decimal digits, nor do share any decimal digit, even if k^m might possibly be expressed in more than one way for some value, e....
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### How close is an odd power of two to a perfect square?

Let $n \ge 1$ be an odd natural number. Define $$f(n)=\min \{\,\, |k| \,\,\, | \, k+2^n \,\,\,\text{is a square}\,\,,k \in \mathbb{Z}\}.$$ That is $f(n)$ measures how close is the power $2^n$ to a ...
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### Self-powers residues

Let $a$, $x$ and $n$ three positive integers such that $\gcd(a,n)=1$ and $x^x=a \mod n^n$. Prove that we can find a positive integer $y$ such that $y^y=a\mod n^{n^n}$. This is what I managed to prove ... 1 vote
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### Whole number solutions to $2n^4+1=m^2$. [duplicate]

What are the whole numbers for which two times the forth power of it plus one is a square? In mathematical notation: $$2n^4+1=m^2;m,n\in\mathbb{Z}$$ My Observations: because of the squares, all ...
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### Fixed point power function for fractional values

I am using below Matlab code to calculate power function i.e. without using built-in function. My requirement is - What improvement/ suggestion make the below function support fractional base and ...
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1 vote
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### Is it the case that for every $n$, $m$, there is an integer expressible as a sum of $m$ distinct positive integer squares in $n$ distinct ways?

I am not a mathematician, but I learn from various sources that we know that, for every n, there is an integer that can be expressed as a sum of two positive integer squares in n distinct ways, and ...
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### Find all positive integers $x,y$ such that $(x+y)(xy+1)$ is a power of $2$ [duplicate]

Find all positive integers $x,y$ such that $(x+y)(xy+1)$ is a power of $2$ It is clear that as we are given a factorisation of the power of two, both of those terms have to be powers of two. $x+y$ is ...
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### Find all triples $(a,b,c)$ giving $3$ powers of $3$ simultaneously [closed]

For which triples $(a,b,c)$ of positive integers are $2a+b,2b+c,2c+a$ all powers of $3$ ? I found out that $a, b, c$ have the same remainder after division by $3$.
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### Infinitely many $n$ which divide $3^{6n-1} - 1$

Are there infinitely many positive integers $n$ such that $n$ divides $3^{6n-1} - 1$? I guess the answer should be positive and the only reasonable idea I have is to consider $n = 3^k - 1$ since if $k$...
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### Are there solutions to $a^5=b^5+c^5+d^5$ where $a,b,c,d \in \mathbb{N}$?

This is a generalization of Fermat's last theorem which states that there are no solutions to: $$a^n=b^n+c^n : a,b,c,n \in \mathbb{N}, n>2$$ A generalization of Fermat's last theorem, Euler powers ...
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### Proof verification of a question associated with perfect cubes

(Poland) For positive integers $a \leq b$, do the following items: (a) Show that $b^3 < b^3 + 6ab + 1 < (b+2)^3$. (b) Find all such a and b for which both $a^3 + 6ab +1$ and $b^3 + 6ab + 1$ are ...
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### Are there any primitive examples besides 56,65 where the sum of a number's cube and its digital reverse's cubes is square?

I saw a question a while back which asked (paraphrased): how many examples exist of the form $a^n + b^n = c^2$ where $b$ is $a$'s digits in reverse, $n \geq 2$, and $a,b,c$ are coprime. For $n = 2$, ...
### Are there positive integers $a,b,c$ such that both $2a(b^c+1),2a(b^c-1)$ are perfect powers?
Are there positive intgers $a,b,c$ such that both $2a(b^c+1),2a(b^c-1)$ are perfect powers? My research We have $2a(b^c+1)-2a(b^c-1)=4a$, so I looked at perfect powers which have a multiple of $4$ as ...
### Does every odd prime power divide a number of the form $\ n^{n-1}+n-1\$?
A conjecture motivated by a factoring project : For every odd prime power $\ P\$ , there is an integer $\ n>1\$ such that $\ P\mid n^{n-1}+n-1\$. In other words , every odd prime power divides ...