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.).

3
votes
1answer
56 views

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. $$2^x-1=3^y.$$
7
votes
0answers
103 views

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 ...
2
votes
0answers
58 views

$A^X+B^Y=C^Z\pm 1$ 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 $A,B,C,N\in \Bbb{N}\space |\space N\ge 3$ ...
0
votes
0answers
54 views

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 ...
0
votes
0answers
14 views

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 =...
7
votes
2answers
106 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 ...
10
votes
2answers
801 views

Diophantine equation $15^x+8^y=17^z$

How to solve the following equation over the integers? $$15^x+8^y=17^z$$ I know that only solution is $(x,y,z) = (2,2,2)$, but how to prove this?
7
votes
4answers
167 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 $m,n$...
6
votes
4answers
880 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 ...
16
votes
2answers
5k views

$\gcd(b^x - 1, b^y - 1, b^ z- 1,…) = b^{\gcd(x, y, z,…)} -1$ [duplicate]

Possible Duplicate: Number theory proving question? Dear friends, Since $b$, $x$, $y$, $z$, $\ldots$ are integers greater than 1, how can we prove that $$ \gcd (b ^ x - 1, b ^ y - 1, b ^ z - 1 ,...
1
vote
0answers
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 ...
6
votes
1answer
158 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 ...
0
votes
2answers
53 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 ...
1
vote
0answers
59 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 ...
10
votes
2answers
235 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 ...
0
votes
2answers
129 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
1
vote
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 ...
1
vote
1answer
67 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 ...
2
votes
0answers
55 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 ...
5
votes
3answers
201 views

If $a^b=c^d$, then $c$ and $a$ are powers of the same number?

I want to know in which situations two numbers that can be expressed as powers can be equal. I think it's intuitive that if two powers (say $a^b$ and $c^d$) are equal, then the bases must be ...
30
votes
6answers
5k views

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 ...
3
votes
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=...
2
votes
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 ...
0
votes
2answers
149 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 ...
2
votes
1answer
64 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 ...
1
vote
0answers
25 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 ...
0
votes
2answers
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 ...
6
votes
2answers
137 views

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+...
4
votes
0answers
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 ...
2
votes
0answers
92 views

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 ...
3
votes
2answers
157 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 ...
-1
votes
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 ...
10
votes
1answer
1k views

To find all $n$ such that $(n-1)!+1$ is a perfect power of $n$

How to find all positive integers $n$ such that $(n-1)!+1$ can be written as $n^k , k\in \mathbb Z^+$ ?
2
votes
0answers
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 $48$ ...
7
votes
0answers
154 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 ...
2
votes
1answer
236 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?
14
votes
2answers
385 views

$x^2+x+1$ is the cube of a prime.

Please help me find all natural numbers $x$ so that $x^2+x+1$ is the cube of a prime number.(Used in here)
2
votes
2answers
166 views

Find all primes $p$ such that $p^2-p+1$ is a perfect cube

Find all primes $p$ such that $p^2-p+1$ is a perfect cube. I found out that p is of the form $18n+1$ and $p=19$ is a solution but I am not getting anything further. $p^2-p-(m^3-1)=0$ $1+4(m^3-1)=k^...
1
vote
0answers
59 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 $(2,1,3)$ ...
9
votes
0answers
117 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=...
1
vote
0answers
63 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 ...
5
votes
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 : ...
1
vote
1answer
569 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 ...
5
votes
0answers
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 ...
3
votes
1answer
265 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?
5
votes
0answers
83 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),(...
0
votes
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}=...
3
votes
0answers
45 views

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 $a^b$ ...
4
votes
2answers
405 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 ...
1
vote
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 ...