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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Lower bound of $n$ th Taxi-cab number $N = a^3 + b^3 = x^3 + y^3$

Let $N,a,b,x,y$ be distinct positive integers such that $$N = a^3 + b^3 = x^3 + y^3$$ Also known as Taxicab numbers or Taxi-cab numbers. see also : https://oeis.org/A001235 let $t(n)$ be the $n$ th ...
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Divisibility by 8 of two summands for Euler hypothesis for fourth power [closed]

Sometimes, I check modular properties of solutions for Diophantine equations. I managed to verify many already-known properties for elements of perfect cuboid. Now, I am exploring restrictions for $a$,...
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Diophantine taxi numbers : $n = p q r = a^3 + b^3 = c^3 + d^3$ for primes $p,q,r$ (example $1729$)

There is something special about $1729$. $$1729 = 10^3 + 9^3 = 12^3 + 1^3$$ It is the sum of two positive integer cubes in exactly two ways. (In fact it is the smallest such integer, but that is not ...
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Perfect cubes with digit-average at least $7.5$

I found so far the following perfect cubes with a digit-average (in base $10$) with at least $7.5$ : ...
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Except for $1$, are there any perfect powers in the sequence $1,122,122333,\cdots,122333\cdots\underbrace{nnn\cdots n}_{n \text{ times}}$?

Other than $1$, is there any perfect powers in the sequence $1,122,122333,\cdots,122333\cdots\underbrace{nnn\cdots n}_{n \text{ times}}$? This is sequence A098129. We certainly know that this is not a ...
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Finding all prime powers of the form $\frac{n(n-1)}{10}$

I was trying to come up with a problem where the solutions would be the numbers 5, 6, 10 and 11. It seems like "finding all integers n, where $\frac{n(n-1)}{10}$ is a prime power", is a good ...
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Simpler solution of the Diophantine equation $x^6-y^6=z^2$.

In the course of answering another question here (link) I struggled quite hard at a rather unsatisfactory argument to show that $$x^6-y^6=z^2,$$ has no nontrivial integral solutions. For that question ...
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Consecutive numbers of the form $n^m + m^{n + 1}$

Playing around with some numbers, I recently noticed that $2024 = 2^{10} + 10^3$ and $2025 = 3^6 + 6^4$. Are there more examples of positive integers $k$ such that both $k$ and $k + 1$ can be ...
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Can the sum of $2025$ consecutive factorials be a perfect power?

Can the sum of $2025$ consecutive factorials be a perfect power? My thoughts: If the sum of $2025$ consecutive factorials is a perfect power, then the number of trailing zeroes in the sum must be a ...
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when could $5^m+5^n$ represented as sum of two squares [closed]

How can we prove that $5^m+5^n$ could be expressed as a sum of two squares if and only if $m-n$ is even with $m,n\in\mathbb{Z}_{>0}$ I was able to prove that any power of $5$ could be expressed as ...
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Proving existence of powers of 10 in a sequence

I have been given the sequence of numbers $$a(n) = 2^{2n-1}-n$$ and I want to know if any of its members is a power of 10 (other than 1). The only thing I've figured out is that the $n$ for such a ...
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Does $(x^2)(x^3) \neq 0$ imply $(x)(x^3) \neq 0$?

Consider a commutative unital algebra $A$ of finite dimension $n>3$ over the reals. The product is defined such that elements are generated with real number coefficients $(a_0, \dotsc, a_n)$ for ...
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Except for $1$, are there any perfect powers in the sequence $1,21,321,4321,\cdots$?

Except for $1$, are there any perfect powers in the sequence $1,21,321,4321,\cdots$? This is sequence OEIS A000422, the concatenation of positive integers from $n$ down to $1$. If there is any perfect ...
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Does there exist a positive integer value of $n$ such that the sum of $1!^n+2!^n+3!^n+\cdots+2024!^n$ is a perfect power?

Does there exist a positive integer value of $n$ such that the sum of $1!^n+2!^n+3!^n+\cdots+2024!^n$ is a perfect power? I know that $n\neq1$, as the last digit of the sum when $n\geq4$ is $3$ and ...
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Is this 3D algebra $T$ power-associative?

Before reading this question it is essential that you understand power associativity https://en.wikipedia.org/wiki/Power_associativity In particular a commutative algebra does not necc imply a power-...
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Faulhaber , primes and irreducible : $S_p(n)=\sum_{k=1}^nk^p = n^a (n+1)^b F_p(n)$

INTRO Let $d\in{\bf N}$ and $n\in{\bf N}^+$ and denote by $S_d(n)$ the sum of the $d$-powers of integers $1$ to $n$: $$S_d(n)=\sum_{k=1}^nk^d$$ A simple use of the binomial theorem in developping the ...
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The Diophantine equation $A^{15} + B^5 + C^5 + D^5 = E^{10}$

Consider the diophantine equation For distinct strict positive integers $A,B,C,D,E$ : $$A^{15} + B^5 + C^5 + D^5 = E^{10}$$ One solution is $$3^{15} + 84^5 + 110^5 + 133^5 = 12^{10}$$ Are there any ...
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Can $1\underbrace{44\cdots4}_{n\text{ times}}$ be a perfect power in other bases?

Can $1\underbrace{44\cdots4}_{n\text{ times}}$ be a perfect power in other bases? Inspired from this question , I know that $144$ and $1444$ are the only perfect powers in base ten, but in other bases,...
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For which $x,n$ is the finite geometric series a perfect power?

Let $x,n,y,q$ be integers greater than one with $n>2$ as well (this is to avoid trivial solutions). The closed form for the geometric series $S(x,n)=\sum_{k=0}^{n-1} x^k$ is $\frac{x^{n}-1}{x-1}$. ...
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Is $7$ the largest possible value of $n$ for which $(1!+2!+3!+…n!)+16$ is a perfect power?

Is $7$ the largest possible value of $n$ for which $(1!+2!+3!+…n!)+16$ is a perfect power? I noticed that $(7!+6!+5!+4!+3!+2!+1!)+16=77^2$ is a perfect power, and I don’t know if that is the largest ...
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Can $(5^k+1)(2^k)$ be a perfect power?

Can $(5^k+1)(2^k)$ be a perfect power? I noticed that $(5^k+1)(2^k)=(10^k+2^k)$, and this cannot be a perfect square since the last digit of $(5^k+1)(2^k)$ is either $2$, or $8$ when $k$ is odd, and ...
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Let $n \in \mathbb{Z}$. Prove $n^6 = k^9 = m^4$ and Let $a = n^6$ $\iff$ $a$ must be a perfect $36$-th power. [duplicate]

Prove $n^6 = k^9 = m^4$ and Let $a = n^6$ $\iff$ $a$ must be a perfect $36$-th power.
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Is there any perfect power in the sequence $12,123,1234,12345,...$?

Inspired from the question Is there any perfect square in the sequence $12,123,1234,12345,...$?, there is no perfect square other than $1$ in the sequence of Smarandache numbers. But I wonder if are ...
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Specific Prime Numbers Chain proofs of maximum length limits.

Considering Prime Numbers $p\in P$ such that $(p^2+4)\in P, (p^2+4)^2+4\in P, ((p^2+4)^2+4)^2+4\in P, (((p^2+4)^2+4)^2+4)^2+4\in P,((((p^2+4)^2+4)^2+4)^2+4)^2+4\in P$, seems to be either very rare, or ...
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The numbers from $11111$ to $99999$ are written in a random order, one after another, forming a single number. Prove that it cannot be a power of $2$.

I found the following problem in a book with advanced math problems for 6th grade that I cannot solve: All the numbers from $11111$ to $99999$ are written in a random order, one after another, thus ...
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When is $63\cdot2^x\pm1$ a perfect power?

When is $63\cdot2^x\pm1$ a perfect power? I noticed that: $63\cdot2^0+1=2^6$ $63\cdot2^1-1=5^3$ $63\cdot2^8+1=127^2$ Other than $0,1,8$, are there any integer values of $x$ such that $63\cdot2^x\pm1$ ...
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Can $1!^2+2!^2+3!^2+\dots+n!^2$ be a perfect power when $n\geq2$?

I know that $S_n:=1!^2+2!^2+3!^2+\dots+n!^2$ cannot be a perfect square because it is equal to $2\pmod{3}$ and it is never a perfect cube because it is equal to $5\pmod{9}$, but can $S_n$ be a higher ...
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Can we always find arbitrarily close powers of $a$ and $b$?

This post was motivated by this other post. I'm aware of Pillai's conjecture (yet to be proven) which states that the gaps in the sequence of perfect powers tend to infinity. However, what happens if ...
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How to prove that $N!+p(N)$ can't be a perfect power, where $p(N)$ is the $N^{th}$ prime?

I've attempted to solve this problem. I can only conclude that if $N!+p(N)=x^y$, since $p(N)>N$, $x$ must be formed by primes greater than $N$ except for $p(N)$. I don't know how to go on. By ...
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Is there a product of $k$ consecutive integers which is one more (or less) than a perfect cube?

I've seen this question with numerous proofs on the product of four consecutive integers being one less than a perfect square. Then I've started to wonder: how many consecutive integers does one need ...
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Can $n!+16$ be a perfect square when $n\geq5$?

Can $n!+16$ be a perfect square? I think $n!+16$ can be a perfect square, since $n!+16$ is $0 \mod 4$, and always $1 \mod 3$ ( when n is $> 5$), and always $5 \mod 11$ ( when $n$ is $> 11$). But ...
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$(a^n-1)(b^n-1)$ can't be a square for all $n$ unless $ab$ is a square

Let $a,b$ be positive integers $>1$ such that $(a^n-1)(b^n-1)$ is a square for all $n\ge 1$. Prove that $ab$ is a perfect square. I'm not asking for a solution to this problem because I already ...
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Is there an analogue to Bertrand's Postulate for primes for perfect powers?

Is there an analogue to Bertrand's Postulate for primes for perfect powers? Bertrand's Postulate: $\forall x \gt 1 \in \mathbb{Z}, \exists p \in \mathbb{P}$ such that $x < p < 2x$. For perfect ...
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Finding the smallest perfect power in a range

Given a range $[a, b]$, how does one find the smallest perfect power contained in the range? i.e., find $a \le k^m \le b$ where $a, b \in \mathbb{R_{\ge 0}}$ and $k, m \in \mathbb{Z_{\ge 0}}, m \ge 2$....
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How to make graphs that avoid certain sub-graphs and have as many edges as possible?

The other day, I needed to create a graph with $10$ vertices, $16$ edges, and which didn't have a loop of four vertices, $C_4$, or a daisy-chain of three $3$-cliques (see picture) as sub-graphs. Turns ...
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Potential solution to Sloane's "Powers of 2" problem?

Neil Sloane gave a problem in a recent Numberphile video here. It seems like there's a solution. But Sloane said it's unsolved and very hard so maybe not. But I'll try to outline the idea for the ...
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Can $5^n+6^n+10^n$ be a perfect power?
Related to this question: Can $f(n)=5^n+6^n+10^n$ with a non-negative integer $n$ be a perfect power ? It is not a perfect power for $0\le n\le 10^5$ Analysis modulo $3$ reveals that a perfect ...
The Problem I've been curious about finding 3 distinct Gaussian integers with equal norm whose imaginary parts cancel out. So $\alpha, \beta, \gamma\in\mathbb{Z}[i]$ with $|\alpha|=|\beta|=|\gamma|$ ...