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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How to prove real Powers of 1 is equal to 1 [closed]

I know that 1^(0.5) = 1 because square root of 1 is 1 . Similarly , I can prove that 1^(x) = 1 * 1 * 1 * 1 * 1 * 1 * ........(x times) = 1 ; where x is an integer like .....-2, -1,0,1,2,3,4..... But I ...
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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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3 answers
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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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5 votes
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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 ...
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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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3 votes
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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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2 answers
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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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Why is it that $(n+(n+1)^2+(n+2)^3) \bmod 144 =0$ starting with $n=3$ and incrementing $n=n+6$?

The series involves: $(n+(n+1)^2+(n+2)^3) \bmod 144 =0$ starting with $n=3$ and incrementing $n=n+6$ Examples: $3+4^2+5^3 = 3+16+125 =144$ and $144 / 144 =1$ $9+10^2+11^3 = 9+100+1331 =1440$ and $1440 ...
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Why is the following for $B= A^x -(A-1)^x-(A-2)^x-(A-3)^x...(A-n)^x$ as long as $B ≥ 0$?

As a web developer that programs in PHP, on my free time I like to run some math algorithms to see the wonder of math. In this experiment, I wanted to check the final results for $B= A^x -(A-1)^x-(A-2)...
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14 votes
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Solutions to $A^N+B^N=C^N \pm 1$

Is there a solution to $A^N+B^N=C^N \pm 1$ where $A,B,C,N\in\Bbb{N}$, such that $A,B,C > 1,N \geq 4$ and $gcd(A,B)=gcd(B,C)=gcd(A,C)=1$? This question was inspired by this one: $A^X+B^Y=C^Z\pm 1$ ...
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2 votes
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Power Tower and Finite Tetration

Let $$\underbrace{a^{a^{.^{.^{.}}}}}_{2020} =2020$$ How can I solve for $a$? Had this been an infinite power tower, I could have used the self-replication. But the finite number of terms (i.e. $2020$ ...
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When do we have $(x y)^2 = x^2 y^2 $?

I just started thinking about algebra so this might be a trivial question. Anyway, Under what conditions do we have $$(x y)^2 = x^2 y^2 $$ ? Does it need to be a group ? Or a groupoid ? Or a monoid ? ...
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Are there more perfect powers of this form?

Let $a,b$ be integers , $a\ge 3$ , $b\ge 1$ If $f_a$ is the $a$ th cyclotomic polynomial, are there any perfect powers of the form $f_a(b)$ apart from $121$ and $343$ ? In the range $$3\le a\le 1\ ...
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4 answers
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Number comparison: $5^{152}<2^{353}$ and $2^{1413}<3\cdot 5^{608}$

Is it possible to prove that $5^{152}<2^{353}$ and $2^{1413}<3\cdot 5^{608}$ without using a calculator or logarithms (middle school math only recommended)? My idea for the first one was to use ...
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Finding remainder using Fermat's Little Theorm

Use Fermat's Little Theorem to find the remainder of $5^{15}$ divided by $1337$. I know Fermat's Little Theorem, but failed to understand how to use it in this. output.944
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1 answer
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Two questions about a special function

Define $$f(n):=\sum_{j=1}^n j^{n+1-j}=1+2^{n-1}+3^{n-2}+\cdots+(n-1)^2+n$$ for a positive integer $n$. With PARI/GP, this function can be calculated with the self-defined function ...
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2 votes
1 answer
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$m^n$ has exactly $9$ digits, all of these digits are distinct.

I was reading the following problem in a book: $2^{29}$ has exactly $9$ digits, all of these digits are distinct, what is the missing digit? I tried to solve it, and I did it correctly. Suddenly, the ...
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2 answers
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Solving $\tau^4 +1=u^2 +v^2$ over the integers

I'm trying to show, that the solutions $t_i$ from Pell's equation $t^2-2s^2=\pm 1$ are not squares (only for the trivial cases). For the even solutions $t_{2i}$ (solutions for the + sign) this is easy,...
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Perfect powers in Horadam sequence

The Horadam sequence $\{W_n\}$ is defined by the Binet formula [Using notation from here]: $$W_n=\frac{A\alpha^n-B\beta^n}{\alpha-\beta}$$ where, $$A\ =\ b\ -\ a\beta \text{ and } B\ =\ b\ -a\alpha$$ $...
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$\lfloor x^n\rfloor\lfloor y^n\rfloor$ is a perfect square

Let $x,y\ge 1$ be non-integer real numbers such that $\lfloor x^n\rfloor\lfloor y^n\rfloor$ is a perfect square for all natural numbers $n$. Does it follow that $x=y$? From this question we know the ...
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Has this family of Diophantine equations ever been studied?

This might be a trivial question, so, if it is, I will delete it as soon as I get an answer. I try to come up with conjectures once in a while. After all, the sleep of reason produces monsters (...
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4 votes
0 answers
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Infinitely many consecutive powers with no consecutive equal digits?

Let $P_k$ be the set of all $n\in\mathbb N$ such that $n^1,n^2,\dots,n^k$ have no consecutive equal digits. In other words, their decimal expansion does not contain '$00$', '$11$' ,'$22$', $\dots$, '$...
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Extending Given Digits to make Perfect Powers

I read this article titled Extending Given Digits to make Primes or Perfect Powers by Sury B, which appeared in the Resonance periodical (October 2010, Indian Statistical Institute, Bangalore). In ...
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6 votes
2 answers
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Power values of polynomial

$f(x) = a_0 + a_1 x + a_2 x^2 + \dots + a_n x^n$ is a polynomial of degree $n$ with positive integer coefficients. Primary problem statement: Is the Exponential Diophantine Equation $f(f(a) + 1) = y^...
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How to check if a number can be represented as the sum of two consecutive perfect cubes.

How to check if a number can be represented as the sum of two consecutive perfect cubes. for eg N = 35 can be represented as the sum of two consecutive perfect cubes 23 and 33
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If there is a way to express an integer as a sum of three non-zero cubes, then there are infinitely many ways

How to prove the following statement: "If an integer can be expressed as a sum of three non-zero cubes in a way, then it can be expressed as a sum of three non-zero cubes in infinitely many ways.&...
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2 votes
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Can a multiperfect number be a perfect power?

(Note: The following post is an offshoot of this earlier MSE question: Can a multiperfect number be a perfect square?.) Let $\sigma(x)$ denote the classical sum of divisors of the positive integer $x$...
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9 votes
3 answers
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Find all positive integers $n$ for which $1372n^4 - 3 $ is an odd perfect square.

Find all positive integers $n$ for which $1372\,n^4 - 3$ is an odd perfect square. I tried $\bmod ,4,5,7$ and failed. Next, I used Vieta’s Theorem and failed again. Any hints, please. Thank you very ...
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9 votes
1 answer
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Find all $n$ which $7(n^2 + n + 1)$ is perfect $4^{th}$ power.

Find all positive integer $n$ , which $7(n^2 + n + 1)$ is perfect $4^{th}$ power. What I tried Let $7(n^2 + n + 1) = a^4$ $\to$ $ 7 | a$ and $a$ is odd. We then get $(n^2 + n + 1) = 343k^4$ ; $k \in \...
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12 votes
1 answer
187 views

Can you find a function $\beta(x)$ where if $a+b=n^m$ then $\beta(\frac{a}{b})$ is irrational?

Can you find a function $\beta(x)$ where if $a+b=n^m$ then $\beta(\frac{a}{b})$ is irrational but if $a+b$ isn't equal to $n^m$ then it is rational ($a$ and $b$ are co-prime)? $n>0$ and $m>1$ $m$...
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Finding an infinite sequence of natural numbers for which any finite partial sum avoids being a perfect power.

$\textbf{Question:}$Is there an infinite set of positive integers such that no matter how we choose some elements of this set, their sum is not a perfect power? $\textbf{My progress:}$I thought of ...
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5 votes
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Techniques for solving an equation $f(x)=y^n$ over the integers?

Despite of the negative result of Hilbert's tenth problem, diophantine equations of the form $$f(x)=y^n$$ can often be fully solved. Here, $f(x)$ is a polynomial with integer coefficients and $n>1$ ...
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6 votes
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Perfect powers nearest to factorials

Suppose, $n\ge 8$ is an integer. Let $s$ be the smallest non-negative integer such that $n!-s$ is a perfect power. Let $t$ be the smallest non-negative integer such that $n!+t$ is a perfect power. ...
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2 votes
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Is there another perfect power?

This is related to this question : Can we conclude $n=p-1$? For which positive integers $\ n\ $ is $$(2n)!+n!+1$$ a perfect power ? For $\ n=1\ $ and $\ n = 2\ $ , we have a perfect power and there is ...
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0 votes
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Closest sums of pairs of reciprocals of prime numbers

This question is related to an upper bound for a problem from my five-year-old question. Let $n\ge 3$ be a natural number and $A_n$ be a set consisting of $1$ and first $n-1$ prime numbers. We want to ...
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