# 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.).

184 questions
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### Counting Perfect Powers

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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### How many ABC three digit numbers are there such that (A+B)^C has three digits and is a power of 2?

I am about to give an exam soon for joining an algorithms course so they sent me a sample test just so I know what's it gonna like and I couldn't solve this problem. The question is the following: ...
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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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### 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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### 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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### 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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### 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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### 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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### finding perfect power factors of an integer

I'm writing a program that does symbolic algebraic computation. The desired behavior for taking the $n$th root of an integer $r$ is to return $p\sqrt[n]{\frac{r}{p^n}}$, where $p$ is the largest ...
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### Solving $x=2^y\cdot z$

For $x$, a known even number that is superior than 2, $y$ an unknown number and $z$ an unknown odd number, I am trying to solve $x=2^y\cdot z$ by factoring powers of 2 from $x$ but I don't really know ...
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### Conversion of arbitrary based numbers to different arbitrary base without division.

I'm attempting to find a better way of converting arbitrary based numbers. I've been taught to first convert the number, lets say 235 (base7)to its equivalent in ...
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### Can $2^s+3^s+\cdots +q^s$ , $q$ odd prime , $s\ge 2$ be a perfect power?

Let $q$ be an odd prime and $s\ge 2$ be an integer. Define the sum $$S(q,s):=\sum_{p\ prime,p\le q} p^s=2^s+3^s+\cdots +q^s$$ Can $S(q,s)$ be a perfect power ? Among other searches, I searched for ...
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### Why $9k \pm 4$ can't be written as sum of four cubes

All integers except possibly those of the form $9k \pm 4$ can be written as the sum of 4 cubes. Are there any heuristics that suggest this is impossible in general, or is it the case that no one ...
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### For any integer n, $78n^2+72n+2018$ is expressed as sum of four perfect cubes. [closed]

For any integer n, $$78n^2+72n+2018$$ is expressed as sum of four perfect cubes. Is this possible that any integer can fit into this expression with 4 perfect cubes?
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### $X^6 + Y^6 + Z^6 = A^6 + B^6 + C^6 = I^6 + J^6 + K^6 = T$

Consider the diophantine equation where all variables are positive and distinct : $$X^6 + Y^6 + Z^6 = A^6 + B^6 + C^6 = I^6 + J^6 + K^6 = T$$ And $T$ is not of the form $V W^6$ for $W>1$. What ...
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### Sums of $5$th and $7$th powers of natural numbers: $\sum\limits_{i=1}^n i^5+i^7=2\left( \sum\limits_{i=1}^ni\right)^4$?

Consider the following: $$(1^5+2^5)+(1^7+2^7)=2(1+2)^4$$ $$(1^5+2^5+3^5)+(1^7+2^7+3^7)=2(1+2+3)^4$$ $$(1^5+2^5+3^5+4^5)+(1^7+2^7+3^7+4^7)=2(1+2+3+4)^4$$ In General is it true for further increase ...
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### Power-of-two integer roundup maths algorithms

Why does $ROUNDUP(x, pow2) = ((x + (pow2 - 1)) \ \&$ ~$(pow2 - 1))$ round up to the next integer in pow (where pow2 must be a power of two ) ? For example, if x = 13, pow = 4 or pow2 = 4*4 = 16 , ...
Let $n$ be the smallest positive integer such that $mn$ is a perfect $k$th power of an integer for some $k \ge 2$, where $m=2^{1980} \cdot 3^{384} \cdot 5^{1694} \cdot 7^{343}$. What is $n+k$? I'm ...
Here is my little research: $4=2^2$ $8=2^3$ $9=3^2$ $12=2^3+2^2$ $13=3^2+2^2$ $16=2^3+2^3$ $17=3^2+2^2+2^2$ $18=3^2+3^2$ $20=2^3+2^3+2^2$ $21=2^3+3^2+2^2$ $24=2^3+2^3+2^3$ and it seems ...