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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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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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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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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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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How can we prove that $3t$ cannot be a perfect cube for any integer $t$ except 9?

If $t \in \mathbb{Z}$ then prove that $3t$ can never be a perfect cube except for $t=9$. How can we prove things like these? I’m pretty new to Number-Theory and I find it difficult to prove things ...
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There exist $n$ different integers in the interval $\big(k^n,(k+1)^n\big)$ whose product is a perfect $n$-th power.

Given a positive integer $ n> 2 $. Prove that there exists a natural number $ K $ such that for all integers $ k \ge K $ on the open interval $ \big({{k} ^{n}}, \ {{(k + 1)} ^{n}}\big) $ there are $...
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How to find the sum of the remainders of a number modulo all the perfect powers less than it?

Perfect powers are numbers of the form $ x^y $ where $x\geq1$ and $y>1$. For example, perfect powers are $\{1, 4, 8, 9, 16, 25, 27, 32, \cdots \}$. If $n$ is a natural number, then we denote by $P(...
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Sum of perfect powers of n natural numbers

The perfect power of an integer $n = p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_r^{\alpha_r}$ are $p_1^2, p_1^3\cdots p_1^{\alpha_1},p_2^2, p_2^3 \cdots p_2^{\alpha_2},(p_1 p_2)^2, (p_2 p_3)^2\cdots$ ...
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For any $k \gt 3$, if $n!+k$ is a perfect power then does there exist any $n\gt k$?

A while ago, I asked a similar question For any $k \gt 1$, if $n!+k$ is a square then will $n \le k$ always be true? where users mathworker21 and WE Tutorial School proved that for non-square $k$, $n\...
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How to find the number of perfect cubes in a range

Can we find the number of perfect cubes in a range. For eg if L = 7 and R = 220 then count will be 5 because 8, 27, 64, 125, 216 are the perfect squares in the range L to R. Can we make a formula ...
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Surprising fact about a certain number-theoretic function

Ante suggested the following function : For natural number $n$ we can observe the $n$ remainders $b_1,...,b_n$ by writing $n$ as $n=a_k \cdot k+b_k$ for $1 \leq k \leq n$ Because of the familiar ...
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How can we find the minimum number by which dividing a number, he number becomes perfect cube?

How can we find the minimum number by which dividing a number, the number becomes perfect cube? eg n = 128 ans = 2(minimum number) because 128 / 2 = 64 which is a perfect cube.
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what are the different ways to check if a number is perfect cube

What are the different ways to check if a number is a perfect cube? I was thinking some ways but want to know different ways to check if a number is a perfect cube or not? And is there any way to ...
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How to find power of a given number?

65536= 4^8------( how can i find the base and power for a number.) Now, if I am randomly given a number. Say, 4096 and I need to find a common base having some power. I mean a^n. Where a is the base ...
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About primes and their powers in bases $\{2,3,4,5,6,7,8,9,10\}$

For some prime $p=p_{10}$, where $p_{10}$ means just that that prime is represented in base $10$, if: *) In at least one base from the set $\{2,3,4,5,6,7,8,9\}$ the number $p^2$ is prime (carefully ...
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Is there for all $N$ an integer that is the sum of, say, twenty $N$-th powers in two different ways?

I believe the following to be true, but I have been unable to find any reference concerning the question. Statement : for every $k\geq 1$, there exists $N = N(k)$ such that there is no non-trivial ...
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$f(x)= \frac{1}{x^{2}-3x-4} $The power series expansion at x = 1.

$f(x)= \frac{1}{x^{2}-3x-4} $ The power series expansion at x = 1. I made an answer, but I am not sure if it is right. $$1/5\left[\sum_{n=0}^∞ (-1)^{(n+1)} -\sum_{n=0}^∞ 1/4^{(n+1)}\right]$$ This is ...
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Odd power of 2 plus even power of 2, is divisible by 6

Prove $2^{2m+1}+2^{2n}$ is divisible by 6 for all integers $n\geq1$ and $m\geq0$. Equivalently one can prove $2+4^k$ is divisible by 3.
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Can we reduce the diophantine equation $x^4+4y^4=z^2$ to $x^4+y^4=z^2$ to show there is no nontrivial solution?

Is it possible to reduce the diophantine equation $$x^4+4y^4=z^2,$$ to the diophantine equation $$x^4+y^4=z^2,$$ to show that there is no nontrivial solution? Please assume the second equation is ...
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Solving Diophantine Equations Involving Exponentials as Positive Integers

Find all positive integers $x$, $y$ and $z$ such that $$x^5+4^y=2013^z$$ Now as $$11|2013 \implies 11|4^y+x^5$$ Now by Euclid's Division Lemma we have $y=5q, 5q+1, 5q+2, 5q+3, 5q+4$, so we get $$4^y \...
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What is the $n^\text{th}$ perfect power, $P(n)$?

Let $P(n)$ denotes the $n^\text{th}$ perfect power of natural numbers (in ascending order without repetition). So, $P(1)=1, P(2)=4, P(3)=8, P(4)=9, P(5)=16, P(6)=25, P(7)=27, P(8)=32, \dots$. Is ...
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Conjecture That The Only Nearly Perfect Numbers Are The Powers Of 2

Call a natural number $n$ “flawed” if its aliquot sum (the sum of its factors less than itself) is $n-1$. Show that the set of flawed numbers is equivalent to the set of perfect powers of $2$. ...
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Pythagorean triple with hypotenuse a power of $2$

Is there a Pythagorean triple whose largest element is a power of 2? That is: are there solutions to $a^2 + b^2 = 2^{2k}$ in the positive integers?
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Find all natural number $n$ for which $3^9+3^{12}+3^{15}+3^n$ is a perfect cube.

Find all natural number $n$ for which $3^9+3^{12}+3^{15}+3^n$ is a perfect cube. What I have tried. $3^9+3^{12}+3^{15}+3^n$ $=3^9(757+3^{n-9})$ Let $757+3^{n-9}=a^3$ Taking modulo $3$: $a^3\...
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Are there forced factors of numbers of this kind?

Let $$f(n):=n^{n^2}+(n+1)^{(n+1)^2}$$ for a positive integer $n$. For clarification $n^{n^2}$ means $n^{(n^2)}$ , analogue for the other summand. Can we find a concrete factor of $\ f(n)\ $ (like ...
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Can $n$ be expressed as $a^b-c^d$, where $a,b,c,d,$ and $n$ are natural numbers, not necessary distinct?

Can $n$ be expressed as $a^b-c^d$, where $a,b,c,d,$ and $n$ are natural numbers, not necessary distinct, and $b$ and $d$ can not be both equal to $1$? For example, when $n=1319$, then $2^{11}-3^6=...
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How many zeros in the decimal representation of $5^n$?

I'm curious about some properties of the powers of 5 $$5^2=25,\quad5^3=125,\quad 5^4=625,\quad 5^5=3125,\quad ...$$ Is it true that at least $50$% of the digits in the decimal representation of $5^n$ ...
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1answer
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Are there nontrivial perfect powers of integers that are nontrivial repdigits?

For example, $6^5=7776$ is close, but not quite a repdigit. Heuristically, it seems to me that there should not be any, because the longer a number with (effectively) random digits is, the less the ...
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If $U, V, W$ and m are natural numbers such that $U^m + V^m = W^m$ , then $m$< max$( U , V ,W)$

If $U, V, W$ and m are natural numbers such that $U^m + V^m = W^m$ , then $m$< max$( U , V ,W)$...How to prove this? It can easily proved that $m$ $\neq$ max$( U , V ,W)$ and $m$ > max$( U , V ...
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Perfect powers of the form $2^n\pm n$?

In the range $$1\le n\le 10^5$$ the only perfect powers of the form $$2^n\pm n$$ are $$2^5-5=3^3$$ and $$2^7-7=11^2$$ How can I prove that there are no more perfect powers of this form ? The case ...
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1answer
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Integer solutions to $2^x-1=3^y$.

Does anybody know how to solve it? I know how to solve linear diophantine equations, but equation like this I've never seen before. $$2^x-1=3^y.$$
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$A^X+B^Y=C^Z\pm 1$ Beal's conjecture “almost” solutions

Beal's conjecture is a generalization of fermat's last theorem. Fermat's last theorem states that there are no solutions to the equation $A^N+B^N=C^N$ where $A,B,C,N\in \Bbb{N}\space |\space N\ge 3$ ...
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Can any Power function (whose base is not zero) , e.g. $2^{n}$, be defined arithmetically (i.e. using addition and multiplication only)? [duplicate]

In other words, I'm looking for a binary relation $P(x,y)$, being arithmetical, i.e being expressed in the first order language of Peano arithmetic (hence non-recursively, i.e. using addition and ...
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Is there any $k$ , for which we can prove that $n^n+k$ is never prime?

Is there any positive integer $k$, such that we can prove that $n^n+k$ is not prime for any positive integer $n$ ? $$n^n+1805$$ has a prime factor not exceeding $43$ up to $n=1805$. However, for the ...
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Method to test if a number is a perfect power?

Is there a general method for testing numbers to see if they are perfect $n$th powers? For example, suppose that I did not know that $121$ was a perfect square. A naive test in a code might be to ...
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On the number of solutions of $pq^m \equiv k\bmod{n}$

It has been shown here that there exist positive integers $n,k$ such that for all primes $p$ and integers $q\ge 1$, $m\ge 2$, we have $$ pq^m \not\equiv k\bmod{n}. $$ $\text{ }$ Hence, let $f(n)$ be ...
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Counting Perfect Powers [closed]

The perfect powers are numbers of the form $x^y$ with $x \geq 1$ and $y>1$. I'm interested in counting the exact number of perfect powers not greater than $N$. I'd like to ask if there's some ...
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1answer
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When does the recurrence $a_k = 3a_{k - 1} + 1$ reach a power of 2?

Almost a year ago, I posed a question on Brilliant involving the following recurrence: Given $n$, let $a_0 = n$ and $a_k = 3a_{k - 1} + 1$. That question asked whether for any choice of $n$, the ...
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Riemann Zeta function, perfect powers, and the Mobius function

I was toying around with the Riemann Zeta function recently and noticed that I could get to a particular representation (valid for $Re(s)>1$) in a couple of different odd ways. The first was by ...
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$n$-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
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Find all positive integers $a, b, c$ such that $21^a+ 28^b= 35^c$ .

Find all positive integers $a, b, c$ such that $21^a+ 28^b= 35^c$. It is clear that the equation can be rewritten as follows: $$ (3 \times 7)^a+(4 \times 7)^b=(5 \times 7)^c $$ If $a=b=c=2$ then ...
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1answer
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2011 IMC Section A Problem 3 [duplicate]

The following is an old competition problem from 2011 IWYMIC: The product $$1!\times 2!\times\cdots 2011!\times 2012!$$ is written on the blackboard. Which factor, in term of a factorial of an ...
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There is no $a, b, n \in N, b \ge 2 $ such $a^{b} = 2017^{n} +43$

I found this problem on some training material for a sixth grade Romanian math competition, and I literally have no clue how to approach it. I am not even sure if my interpretation of the problem text ...
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1answer
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How sure can we be that $\underbrace {k\cdots k}_{m\ k's}$ cannot be a perfect power?

Let $k$ and $m$ be integers , $m\ge 2$ and $k\in [1,9]$. Denote $$n=k\cdot \frac{10^m-1}{9}=\underbrace {k\cdots k}_{m\ k's}$$ As far as I know, it is unknown whether a rep-unit can be a cube. How ...
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Are there infinite many perfect powers consisting of at most two decimal digits?

Suppose, $a,b\ge 2$ are integers. Then $$N=a^b$$ is a perfect power. Assuming that $a$ is not divisible by $10$, are there infinite many such perfect powers consisting of at most $2$ distinct decimal ...
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Quick! Prove that $3^{15}+37$ is a square.

CONTEXT: I have been studying the odd powers of $3$ and trying to determine when they are "close to" square numbers; more specifically, I have conjectured that there exist finitely many solutions $m,n$...
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What is the smallest integer greater than 1 such that $\frac12$ of it is a perfect square and $\frac15$ of it is a perfect fifth power?

What is the smallest integer greater than 1 such that $\frac12$ of it is a perfect square and $\frac15$ of it is a perfect fifth power? I have tried multiplying every perfect square (up to 400 by two ...
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1answer
45 views

Finding values to make an equation with a $x^4$ into a perfect square

For what values of p and q is the expression $$x^4+ 6x^3+ 13x^2+ px + q$$ a perfect square? The answer I got didn't seem correct so if someone break it down to me how about going this problem it would ...
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2answers
501 views

How many perfect powers are there amoung the first 1000 positive integers

A positive integer $n$ is called a perfect power if and only if there exist two positive integers $p,q$, both are greater than 1, such that $n=p^q$. For example $8=2^3, 36=6^2$ are perfect powers, but ...
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224 views

What power of a number is closest to a given integer?

Lets say there is a positive number (integer) given ($N$). What should be the simplest way to find the $K^p$ (where $K$ and $p$ both $>1$) which is closest to $N$ amongst all the possible ...

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