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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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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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44 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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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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Prove that if $\Sigma_{i=1}^{n}(a_i)\le2^{n-4}$ then $a_2!a_3!\dots a_n!+1$ is not a power of $a_1$

The problem statement is Let $a_1,a_2,\dots,a_n$ be a strictly decreasing sequence of positive integers, with $a_1\equiv5\ (\text{mod }8)$. Prove that for all positive integers $n$, if $$a_1+...
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What's equal this:${\frac 12 ^ \frac 13} ^{\cdots \frac1n}$? [duplicate]

I want to know wht's equal this number :${\frac 12 ^ \frac 13} ^{\cdots \frac1n}$, however it's seems trivial that is converge to 1 , but i don't know how i can evaluate it , Really i have two ...
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Are there infinite many pairs of primes $(p/q)$ such that $p^3+q^3$ is a perfect square?

The pairs of primes $(p,q)$ with $p\le q\le 10^5$ such that $p^3+q^3$ is a perfect square , are (the last entry is the maximum possible exponent of the perfect power $p^3+q^3$ which is only different ...
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$2^W = (2a_1+1)^2 \cdot 2^{M_1}-a_1$ - is there a nontrivial diophantine solution in $M_1,a_1$ given possible?

Update/error: I had a small error in the $(2a+1)$-term in the formula which should have been $(a+1)$ . Because I added now my derivations in long and broad at the end of the question I also rewrite a ...
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The construction of a number with given digits [closed]

Given a set of decimal digits. And given a set of primes $\mathbb{P}$, find some $p \in \mathbb{P}$ such that $p^n, n \in \mathbb{N} $ contained in itself all the digites from a given set, and it does ...
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Does concatenating the perfect powers ever lead to a prime?

I search a prime number of the form $$4891625\cdots $$ emerging by concating the perfect powers $p>1$ upto some specific limit $L$ With $L=529$, we get a number splitting into a $31$ and a $48$ ...
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Is $2^9$ the only power of two that is the sum of two odd perfect powers?

Let $m\ge 1$ be an integer. For which $m$ can we find odd perfect powers $a,b$ with $a+b=2^m$ ? The only solution, I found for $m\le 80$ is $m=9$ with the representation $$2^9=13^2+7^3$$ The ...
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167 views

$p^2-p+1$ is a perfect cube of a prime [closed]

Determine with a proof all prime numbers p such that p$^2$-p+1 is a cube of a prime number.By trial and error method 19$^2$-19+1=7$^3$Is it the only p?How should I prove it?
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Can I complete the euclidean case without elliptic-curve-theory?

Can I prove without using properties of the elliptic curves , that the equations $$x^2+y^3=z^6$$ $$x^2+y^6=z^3$$ $$x^3+y^6=z^2$$ have no solution in coprime positive integers except $(2,1,3)$ ...
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Can a prime have arbitary many representations as a sum of two perfect powers?

Let $p$ be a prime number and $f(p)$ be the number of representations $$p=a+b$$ with perfect powers $0<a< b$ For example, $13$ has the only representation $$4+9=13$$ hence $f(13)=1$ $$41=9+32=...
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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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1answer
215 views

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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“Schäffer's conjecture” on equation $1^k+2^k+\cdots+x^k=y^n$

In 1956 J. J. Schäffer proved that the equation $$1^k+2^k+\cdots+x^k=y^n$$ for fixed integers $k\ge1,n\ge2$ has only finitely many solutions in positive integers unless $(k,n)\in\{(1,2),(3,2),(3,4),(...
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127 views

Is this set of solutions complete?

Let $p,q,r$ be distinct primes and $a,b,c\ge 2$ integers. The equation $$p^a+q^b=r^c$$ has the following solutions : $$2^4+3^2=5^2$$ $$2^5+7^2=3^4$$ $$2^2+11^2=5^3$$ $$2^7+17^3=71^2$$ $$7^3+13^2=...
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Which are the solutions of $a^2+b^3=c^6$ with coprime integers $a,b,c$?

Here https://en.wikipedia.org/wiki/Fermat%E2%80%93Catalan_conjecture the fermat catalan-conjecture is mentioned. In the german version, I read that in the case $$\frac{1}{m}+\frac{1}{n}+\frac{1}{k}=...
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Are there only finite many n-tupels of factorials summing up to a non-trivial power?

Let $n\ge 1$ be a positive integer and $S$ be the set of all $n$-tuples of positive integers $1\le a_1\le a_2\le \cdots \le a_n$ such that $$\sum_{j=1}^n a_j!$$ is a non-trivial power (a number $a^b$ ...
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230 views

Number of elements $n \in \{1, …, 100 \}$ such that $n^{4} - 20n^{2} + 100$ is of the form $k^{4}$ with $k$ an integer.

Find the number of elements $n \in \{1, ..., 100 \}$ such that $n^{4} - 20n^{2} + 100$ is not of the form $k^{4}$ with $k$ an integer. Notice that $$ n^{4} - 20n^{2} + 100 = (n^{2} - 10)^{2} $$ We ...
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38 views

Waring’s analogue for fractions?

I was thinking about Waring’s problem and Also about fractions. So Naturally we can combine those. Is Every positive fraction the Sum of at most 4 positive fractions squared ? How about cubes ? Etc ...
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Naturals representable as differences of powers

With paper-and-pencil method I found only a first $5$ cases: $$1=3^2-2^3$$ $$2=3^3-5^2$$ $$3=2^7-5^3$$ $$4=5^3-11^2$$ $$5=2^5-3^3$$ This looks interesting and if a natural $n$ can be represented ...
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Which cycles are possible by repeated summation of the cubes of the digits of a number?

Here : Digital root with squared digits the possible cycles of repeated summation of the squares of the digits of a number are mentioned. What about cubes ? $1$ , $153$ , $370$ , $371$ and $407$ ...
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A Problem of Perfect Powers. Calculating the amount of numbers were $a^b=n$. [closed]

Call an integer $n$ oddly powerful if there exist positive integers $a$ and $b$, where $b>1$, $b$ is odd, and $a^b = n$. How many oddly powerful integers are less than $2010$?
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Is there another prime $p$ such that $S(p)$ is prime?

Denote $$S(p):=2^2+3^3+5^5+\cdots +p^p$$ $S(p)$ is prime for $p=3,7,89$. Is there another prime $p$ such that $S(p)$ is prime ? Is the number of primes $p$ such that $S(p)$ is prime, finite ?
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Density of n-th Power of Rationals in $\mathbb{Q}$

I'm working through Rudin, and was attempting to prove that since $\mathbb{R}$ satisfies the least-upper-bound property, then each positive element of $\mathbb{R}$ has an $n$-th root. I came to ...
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What is the largest number whose first digits can be calculated? [duplicate]

The way to calculate the first digits of $a^b$ is to compute the $log_{10} a$ and multiply by $b$, and find the fractional part. But if there are too many digits in the integer part of the result, ...
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Are there more non-trivial powers of the form $S_p$?

Let $S_p:=2+3+5+\cdots p$ denote the sum of primes up to $p$. I searched for $S_p$ being a non-trivial power and only found the following squares so far : ...
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Solving $x^2+x+1=7^n$

Do simple equations $x^2+x+1=7^n$ have infinitely or finitely many solutions? In fact, what about the general equation $x^2+x+1=p^n$ where $p$ is a prime congruent to $1$ $\pmod 3$? Are there any ...
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1answer
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Do there exist any positive integers $n$ such that $\lfloor{e^n}\rfloor$ is a perfect power? What is the probability that one exists?

I know there are no integers for which the value is specifically a perfect square for the first 73 integers because I plugged the function $\sqrt{\lfloor{e^{\lfloor{x}\rfloor}}\rfloor}-\lfloor{\sqrt{\...
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$ 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 , ...
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An elementary number theory problem

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 ...
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2answers
63 views

numbers representable as sums of perfect powers

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 ...
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1answer
343 views

What modulus should be chosen when proving impossibility of $|2^m-3^n|=N$?

In this question it was asked whether there exist integer solutions to the equation $$|2^m-3^n|=35,$$ and I responded by saying that, modulo $85$, there do not exist any integer solutions to $$2^m ...
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4answers
223 views

Solution to $|2^m-3^n|=35$?

As in a previuos question I'm wondering about equations like $|A^m-B^n|=C$, where $A,B,C\in\mathbb N$ are given and solutions $m,n\in\mathbb N$ are wanted. As it seems there are no known general ...
4
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1answer
68 views

When can we factor $\displaystyle\sum_{l=0}^{n-1} m^l$

Inspired by this question which in the case $k=2$ we would need to factor the second factor too: $$\displaystyle\sum_{l=0}^{n-1} m^l$$ Can we say anything in general for which $n$ or $m$ this will be ...
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0answers
32 views

Primes w.rt. the power operator, are they useful?

The multiplication operator is very important in algebra and number theory. Tied to it is the property of primality. A number that is prime can only be written as a product of 1 and itself and not as ...
6
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2answers
112 views

Is there a prime of the form $a^b+b^c+c^d$ with consecutive primes $a,b,c,d$?

Suppose $a,b,c,d$ are consecutive prime numbers with $a<b<c<d$ Can $$a^b+b^c+c^d$$ be a prime number ? On the one hand, I did not find a prime for $a\le 4723$ (maybe, someone double-...
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1answer
71 views

What is the Sum of the Amount of Powers of Primes $p$ between $n-2p$ and $n$?

What is the sum of the amount powers of primes $p$ between $n-2p$ and $n$, with $n>p$, and$2 < p < \frac{n}{2}$? By powers of primes, I mean an integer of the form $p^k$, not an integer of ...
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1answer
51 views

How many triplets of natural numbers, that satisfy the specific conditions, do there exist?

How many triplets $(x, y, z)$, such that $x, y, z \in \mathbb{N}$, $z > 1$ and $x^2 + 1 = y^z$, do there exist? I failed to find any. Any help will be appreciated.