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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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!...
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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 ...
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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 $...
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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.
user1202451
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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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Congruence equation $v^q \equiv a \pmod p $
I am reading a text on number theory, but I am confused about the following.
Let $a$ be neither $\pm 1$ nor a perfect square. Suppose $h$ is the largest positive integer such that $a$ is a perfect $h$...
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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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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 $...
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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 $\;...
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$(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 ...
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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 $|\alpha|=|\beta|=|\gamma|$ ...
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Formula for working out the greatest power that another number is divisible by
Is there a formula to answer the question
What is the highest value of $n$ where it is true that $b^n$ is a factor of $x$?
where $b$ and $x$ are known positive integers and $n$ is an unknown non-...
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To prove that irrational powers of 1 is equal to 1 [closed]
It is easy to prove that if 1 is raised
to any power of a rational number ( say $x$), then :
$(1)^x = (1)^{p/q}$
(because $x$ can be expressed as $p/q$ where $p$ and $q$ are integers except $q≠0$). ...
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Prove that we can express $n^k$ as a sum of $n$ consecutive odd natural numbers
Prove that $\forall n \in \mathbb{N},$ we can express $n^k$ where $k
\geq2$ is an integer, as a sum of $n$ consecutive odd numbers.
My Solution -
Let $2m+1$ be the first odd number , $m \geq 0 , m \...
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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 ...
user14479
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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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I can't understand this solution with $\equiv$
I found a solution for finding the units digit of a sum. It said that the $\equiv$ means "has the same units digit as". Using this, they said that $3^1+3^3+3^5+...+3^{2009} \equiv 3+7+3+...+...
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About the diophantine equation $x^3+y^3=z^8$
After solving the exercise
The equation $x^n+y^n=z^{n+1}$ has infinite solutions in the positive integers.
I realized that this can be easily generalized to
If $m\equiv 1 \pmod n$, the equation $x^...
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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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Squareful Part of a Family of Numbers
For any integer $x$, let $f(x)=64x^6-27$. Now, write $f(x)=g(x)h(x)$, where $g(x)$ is the squarefree part of $f(x)$ and $h(x)$ is its squarefull (powerful) part. For example, $f(5)=13^2\cdot 61\cdot ...
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Consecutive prime powers that are not prime
I am looking for consecutive entries $(a_k,a_{k+1})$ in the sequence of prime powers
$$(a_n)=(2,3,2^2,5,7,2^3,3^2,11,13,2^4,17,19,23,5^2,3^3,29,31,2^5\cdots)$$ such that neither $a_k$ nor $a_{k+1}$ ...
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Does there exist a positive integer $k$ and an irreducible polynomial $P$ of degree at least $2$ such that $P$ is a power of $k$ infinitely often?
Does there exist a positive integer $n\in\mathbb{Z}$ and an irreducible polynomial $P\in\mathbb{Z}[X]$ of degree at least $2$ such that there are infinitely many pairs of positive integers $(m,k)$ ...
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How well does this method of checking if an integer $N$ is a square perform?
The method is based on the following observation: an integer $N=n^2$ is a square which can also be written as $N=n^2=(a+b)^2=a^2+2ab+b^2$ with:
$a=(n−1)/2$
$b=(n+1)/2$
We need to consider two cases, $...
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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 $n = 2$, ...
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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 ...
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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 ...
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