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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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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 $a$,...
Rators Asepo's user avatar
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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 ...
mick's user avatar
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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 ...
Thirdy Yabata's user avatar
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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 ...
Servaes's user avatar
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Is $k=11$ the largest value of $k$ such that $(\lfloor\sqrt{1!+2!+3!+\cdots+k!}\rfloor+1)^{2} - (1!+2!+3!+\cdots+k!)$ is a perfect power?

Is $k=11$ the largest value of $k$ such that $(\lfloor\sqrt{1!+2!+3!+\cdots+k!}\rfloor+1)^{2} - (1!+2!+3!+\cdots+k!)$ is a perfect power? While I playing on my calculator, I observed the following: $(\...
Thirdy Yabata's user avatar
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Conjecture: $\forall n \geq n_0\exists k \geq 0: \gcd(2^k-1, \frac{p_n\#}{6}) = 1$.

Context & Interest. See this MSE post about a twin-prime related topology. Basically $0$ is a easily seen to be a generic point in this topology. Every generic point is clearly dense as a ...
SeekingAMathGeekGirlfriend's user avatar
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Conjecture on digits in arithmetic progression

Let $n\geq 100$ be some positive integer in base $10$. Take its consecutive digits in groups of $k$ digits each. I have the following Conjecture: If the digits of some positive integer $n\geq 100$ in ...
Juan Moreno's user avatar
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Is there a base $b\geq11$ that we obtain a perfect power after we concatenate the first $n$ positive integers of that base?

Is there a base $b\geq11$ that we obtain a perfect power after we concatenate the first $n$ positive integers of that base? In base ten, the concatenation of first $n$ positive integers does not give ...
Thirdy Yabata's user avatar
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Is $357911$ the only perfect power that is obtained by concatenating $5$ or more consecutive odd numbers(in decimal)?

Is $357911$ the only perfect power that is obtained by concatenating $5$ or more consecutive odd numbers(in decimal)? I noticed that while memorizing all perfect cubes from $1$ to $10^{6}$, $357911=71^...
Thirdy Yabata's user avatar
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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 ...
Richard Wang's user avatar
5 votes
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362 views

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 ...
Thirdy Yabata's user avatar
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2 answers
181 views

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 ...
Mostafa dd's user avatar
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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 ...
SuspiciousGarbage's user avatar
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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 ...
mick's user avatar
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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 ...
Thirdy Yabata's user avatar
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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 ...
Thirdy Yabata's user avatar
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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-...
mick's user avatar
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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 ...
mick's user avatar
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2 votes
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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 ...
mick's user avatar
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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,...
Thirdy Yabata's user avatar
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Are $8, 36, 216,$ and $2304$ the only perfect powers that is $6$ less or more than a primorial?

Are $8, 36, 216,$ and $2304$ the only perfect powers form of $p\#$$\pm6$, where $p\#$ is primorial? I noticed that when I try to add or subtract $6$ from a primorial, the results are: $2+6=8$, which ...
Thirdy Yabata's user avatar
2 votes
3 answers
177 views

Is $1 \times 1 + 2 \times 2 + 3 \times 4 + 4 \times 8 = 49$ a coincidence? (Is $\sum_{i=0}^k(i+1)2^i$ ever a square again?)

When watching a gaming video, I noticed an intriguing fact: $$ 1 \times 1 + 2 \times 2 + 3 \times 4 + 4 \times 8 = 49, $$ which is a square number. I asked myself, is this a coincidence or not? ...
Benjamin Wang's user avatar
5 votes
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301 views

Can $!1+!2+!3+\cdots+!n$ be a perfect power?

Can $!1+!2+!3+\cdots+!n$ be a perfect power if $n\geq3$? Note that $!n$ is a subfactorial. I do know that $1!+2!+3+\cdots+n!$ is only a perfect power if $n=1, 3$, since when $n\geq9, 1!+2!+3!+\cdots+9!...
Thirdy Yabata's user avatar
5 votes
0 answers
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Is $1105$ the only Poulet-number of the form $2^a+3^b$?

Conjecture : $1105$ is the only Poulet-number that can be written in the form $2^a+3^b$ with positive integers $a,b$ A Poulet-number is a composite number $N$ satisfying the congruence $2^{N-1}\equiv ...
Peter's user avatar
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Is $S_n=1!^{1!}+2!^{2!}+3!^{3!}+\cdots+n!^{n!}$ a perfect power if $n\geq2$

Is $S_n=1!^{1!}+2!^{2!}+3!^{3!}+\cdots+n!^{n!}$ a perfect power when $n\geq2$? I know that: $S_n\equiv2\pmod{3}$ if $n\geq2$, so $1!^{1!}+2!^{2!}+3!^{3!}+\cdots+n!^{n!}$ is never a perfect square if $...
Thirdy Yabata's user avatar
6 votes
0 answers
148 views

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 $\frac{x^{n}-1}{x-1}$. ...
Integrand's user avatar
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7 votes
2 answers
227 views

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 ...
Thirdy Yabata's user avatar
2 votes
0 answers
270 views

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 ...
Thirdy Yabata's user avatar
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1 answer
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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 ...
Thirdy Yabata's user avatar
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1 answer
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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 ...
Eugen's user avatar
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6 votes
2 answers
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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 ...
Marius Bancila's user avatar
1 vote
2 answers
230 views

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 $63\cdot2^x\pm1$ ...
Thirdy Yabata's user avatar
36 votes
2 answers
2k views

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 ...
Thirdy Yabata's user avatar
10 votes
2 answers
206 views

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 ...
Alma Arjuna's user avatar
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2 votes
1 answer
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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 ...
Falcon's user avatar
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2 votes
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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 ...
Rusurano's user avatar
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4 votes
1 answer
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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 ...
Thirdy Yabata's user avatar
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Can $300$…$04$ be a perfect square? [duplicate]

$300…04$ is divisible by $4$ and always $1$ mod $3$. Sometimes, $300$…$04$ is $1$ mod $11$, which is a quadratic residue modulo $11$. But when I tried to check if there is any perfect square form of $...
Thirdy Yabata's user avatar
1 vote
4 answers
554 views

Can $\,9\!\cdot\!10^n+4\,$ be a perfect square? [duplicate]

I think $\,9\!\cdot\!10^n+4\,$ can be a perfect square, since it is $0 \pmod 4$ (a quadratic residue modulo $4$), and $1 \pmod 3$ (also a quadratic residue modulo $3$). But when I tried to find if $\;...
Thirdy Yabata's user avatar
3 votes
1 answer
147 views

$(a^n-1)(b^n-1)$ 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 ...
PNT's user avatar
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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 ...
vvg's user avatar
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0 votes
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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 $k, m \in \mathbb{Z_{\ge 0}}, m \ge 2$....
vvg's user avatar
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4 votes
0 answers
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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 ...
Chris Wolird's user avatar
11 votes
2 answers
745 views

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 ...
Chris Wolird's user avatar
11 votes
5 answers
508 views

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 ...
Peter's user avatar
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2 votes
1 answer
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Finding three perfect complex powers with equal norm and cancelling imaginary parts

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 $|\alpha|=|\beta|=|\gamma|$ ...
WhiteStoneJazz's user avatar

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