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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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$\gcd(b^x - 1, b^y - 1, b^ z- 1,…) = b^{\gcd(x, y, z,…)} -1$ [duplicate]

Possible Duplicate: Number theory proving question? Dear friends, Since $b$, $x$, $y$, $z$, $\ldots$ are integers greater than 1, how can we prove that $$ \gcd (b ^ x - 1, b ^ y - 1, b ^ z - 1 ,...
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1answer
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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. $$8,1,15,10,6,3,13,12,4,5,11,14,2,7,9$$ ...
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1answer
1k views

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 $n^k , k\in \mathbb Z^+$ ?
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2answers
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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}}{...
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1answer
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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 ...
8
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6answers
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 ...
6
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2answers
277 views

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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3answers
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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 ...
14
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2answers
385 views

$x^2+x+1$ 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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6answers
484 views

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 ...
1
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1answer
358 views

Solving a little Diophantine equation:$(n-1)!+1=n^m$ [duplicate]

How can I solve this Diophantine equation: $$(n-1)!+1=n^m$$ with $n,m$ positive integers? From Wilson's theorem we can note that $n$ is a prime number. I proved to rewriting the equation as:$$(n-2)!=...
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4answers
229 views

Solution to $|2^m-3^n|=35$?

As in a previuos question I'm wondering about equations like $|A^m-B^n|=C$, where $A,B,C\in\mathbb N$ are given and solutions $m,n\in\mathbb N$ are wanted. As it seems there are no known general ...
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2answers
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What better way to check if a number is a perfect power?

What better way to check if a number is a perfect power? Need to write an algorithm to check if $ n = a^b $ to $ b > 1 $. There is a mathematical formula or function to calculate this? I do not ...
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2answers
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Given an integer, how can I detect the nearest integer perfect power efficiently?

If you give me an integer N, how can I detect the nearest integer perfect power, larger or smaller than N? In other words, the perfect power the distance between N and which is less than the ...
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1answer
592 views

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=...
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2answers
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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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1answer
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What modulus should be chosen when proving impossibility of $|2^m-3^n|=N$?

In this question it was asked whether there exist integer solutions to the equation $$|2^m-3^n|=35,$$ and I responded by saying that, modulo $85$, there do not exist any integer solutions to $$2^m ...
5
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0answers
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Prime candidates of the form $n^{(n^n)}+n^n+1$?

Let $$\large f(n)=n^{(n^n)}+n^n+1$$ Checking $f(n)$ for $2\le n\le 100$, I noticed that $f(n)$ has a small prime factor except for $n=12,53$ and $60$ For $n=53$, I found the prime factor $7074407$ , ...
7
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1answer
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Pairwise sums are perfect squares .

I thought of this problem as a simplification of the Euler-Brick problem . $1)$For which $n$ is it possible to find $n$ distinct positive integers $a_1,a_2,\ldots,a_n$ such that all their pairwise ...
5
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1answer
156 views

The smallest integer whose digit sum is larger than that of its cube?

79 is an example of a number whose digital sum is greater than that of its square (6241). Which is the least number, if any, whose digital sum is greater than that of its cube?
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3answers
751 views

Sums of Fourth Powers

While fooling around on my calculator I found: $$7^4 + 8^4 + (7 + 8)^4 = 2 * 13^4$$ $$11^4 + 24^4 + (11 + 24)^4 = 2 * 31^4$$ I'm intrigued but I can't explain why these two equations are true. Are ...
0
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2answers
128 views

$n$-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
3
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1answer
151 views

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 ...
2
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0answers
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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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0answers
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 ...
3
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0answers
55 views

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 ...
3
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2answers
157 views

$2^W = (2a_1+1)^2 \cdot 2^{M_1}-a_1$ - 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 ...
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0answers
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Shortcuts for next perfect power computation

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 ...
0
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2answers
135 views

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 ...