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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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Simplifying $\gcd((7n+3)a, 2a-b)$, where $a$ and $b$ are perfect powers

As the question states, I want to find $\gcd((7n+3)a, 2a-b)$. Here $a = c^{n-2}$, $b = d^{n-2}$, where $c, d$ are relatively prime and $c > d$, and $n$ is a positive integer greater than 2. Let $x =...
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96 views

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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4answers
850 views

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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31 views

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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52 views

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
156 views

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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53 views

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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116 views

$n$-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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219 views

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
40 views

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
66 views

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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54 views

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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162 views

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
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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132 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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1answer
59 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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24 views

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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42 views

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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147 views

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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2answers
155 views

$2^W = (2a_1+1)^2 \cdot 2^{M_1}-a_1$ - 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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3answers
62 views

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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60 views

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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148 views

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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1answer
224 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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57 views

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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116 views

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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61 views

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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83 views

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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1answer
560 views

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
261 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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0answers
82 views

“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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1answer
131 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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1answer
43 views

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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2answers
397 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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1answer
39 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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2answers
64 views

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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42 views

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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70 views

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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126 views

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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1answer
81 views

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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2answers
49 views

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, ...
5
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0answers
58 views

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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0answers
144 views

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
63 views

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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0answers
85 views

$ 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 ...
10
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5answers
182 views

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 ...