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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.).

69
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2k views

Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$

About a month ago, I got the following : For every positive rational number $r$, there exists a set of four positive integers $(a,b,c,d)$ such that $$r=\frac{a^\color{red}{3}+b^\color{red}{3}}{...
53
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1answer
5k views

Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

I've known that one can arrange all the numbers from $1$ to $\color{red}{15}$ in a row such that the sum of every two adjacent numbers is a perfect square. $$8,1,15,10,6,3,13,12,4,5,11,14,2,7,9$$ ...
38
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4answers
3k views

Is this a new method for finding powers?

Playing with a pencil and paper notebook I noticed the following: $x=1$ $x^3=1$ $x=2$ $x^3=8$ $x=3$ $x^3=27$ $x=4$ $x^3=64$ $64-27 = 37$ $27-8 = 19$ $8-1 = 7$ $19-7=12$ $37-19=18$ $18-...
33
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3answers
1k views

When is $2^n \pm 1$ a perfect power

Is there an easy way of showing that $2^n \pm 1$ is never a perfect power, except for $2^3 + 1 = 3^2 $? I know that Catalan's conjecture (or Mihăilescu's theorem) gives the result directly, but I'm ...
29
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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 ...
21
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1answer
379 views

Integers $n$ for which the digit sum of $n$ exceeds the digit sum of $n^5$

This question is strongly inspired by The smallest integer whose digit sum is larger than that of its cube? by Bernardo Recamán Santos. The number $n=124499$ has digit sum $1+2+4+4+9+9=29$ while its ...
16
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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 ,...
15
votes
1answer
592 views

A $\frac{1}{3}$ Conjecture?

Question: Let $A(n)$ be a finite square $n \times n$ matrix with entries $a_{ij}=1$ if $i+j$ is a perfect power; otherwise equals to $0$. Is it true that $${1 \above 1.5 pt n^2}\sum_{i=1}^n \sum_{j=...
14
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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)
13
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3answers
782 views

Perfect powers of successive naturals: Can you always reach a constant difference?

I was thinking about what happens if you take a sequence of consecutive squares, for example 1,4,9, 16. Taking the differences gives you another sequence, 7,5,3. And taking the differences between ...
12
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6answers
484 views

If $ a + b + c \mid a^2 + b^2 + c^2$ then $ a + b + c \mid a^n + b^n + c^n$ for infinitely many $n$

Let $ a,b,c$ positive integer such that $ a + b + c \mid a^2 + b^2 + c^2$. Show that $ a + b + c \mid a^n + b^n + c^n$ for infinitely many positive integer $ n$. (problem composed by Laurentiu ...
11
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1answer
201 views

Prove that $\sum\limits_{k=1}^n (k!)^2$ is not a perfect square when $n\ge2$

Prove that $\displaystyle \forall n\geq 2, \sum_{k=1}^n (k!)^2$ is never a perfect square. I'm far from well-read in number theory and I can't make any significant progress with this problem. I ...
10
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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 $n^k , k\in \mathbb Z^+$ ?
10
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5answers
184 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 ...
10
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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?
10
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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 ...
9
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2answers
990 views

Given an integer, how can I detect the nearest integer perfect power efficiently?

If you give me an integer N, how can I detect the nearest integer perfect power, larger or smaller than N? In other words, the perfect power the distance between N and which is less than the ...
9
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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=...
8
votes
6answers
758 views

How to compute 2-adic square roots?

I know that, for a $2$-adic unit to be a perfect square, it must be of the form $\cdots001.$, for example the number $17$ ($10001.$) is a $2$-adic square. How would I go about finding the $2$ adic ...
8
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4answers
185 views

Can $\frac{a^4}{b^3}$ be arbitary close to $1$?

I searched for positive integers $a$ and $b$ for which the difference of $a^4$ and $b^3$ is small compared to the powers $a^4$ and $b^3$. The most spectacular example I currently have is $$9825757^4-...
8
votes
2answers
187 views

A conjecture about primes

Conjecture: For each prime $p$ there are an infinite number of primes $q$ such that $p+q$ is a perfect square. I have done a lot of tests using Bigz and I believe that it's possible to prove. I ...
8
votes
1answer
201 views

Difference of fourth powers in three ways

I'd like to find a number that's the difference of fourth powers in three ways or more. I.e.: $$k=a^4-b^4=c^4-d^4=e^4-f^4$$ Is this possible? There seem to be plenty of examples of differences of ...
8
votes
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 ...
7
votes
3answers
751 views

Sums of Fourth Powers

While fooling around on my calculator I found: $$7^4 + 8^4 + (7 + 8)^4 = 2 * 13^4$$ $$11^4 + 24^4 + (11 + 24)^4 = 2 * 31^4$$ I'm intrigued but I can't explain why these two equations are true. Are ...
7
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4answers
395 views

Prove that every number of the sequence $49,4489,444889,\ldots$ is a perfect square [duplicate]

Prove that every number of the sequence $49,4489,444889,\ldots$ is a perfect square. We can observe that $49=7^2, 4489=67^2, 444889=667^2, \ldots$ I have tried expanding terms of the sequence, and ...
7
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3answers
486 views

How do I prove that $3^n - 3$ is never a square number?

Prove that there do not exist positive integers $a$ and $n$ such that $a^2 + 3 = 3^n$. I have tried writing the equation in another way: $a^2 = 3^n - 3 = 3(3^{n-1} -1)$ but that doesn't get me ...
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 $m,n$...
7
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1answer
2k views

Product of three consecutive positive integers is never a perfect power

I am trying to prove that the product of three consecutive positive integers is never a perfect power. Can anyone point to gaps in my proof and/or post an alternate solution? Let the three positive ...
7
votes
2answers
105 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 ...
7
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1answer
664 views

Pairwise sums are perfect squares .

I thought of this problem as a simplification of the Euler-Brick problem . $1)$For which $n$ is it possible to find $n$ distinct positive integers $a_1,a_2,\ldots,a_n$ such that all their pairwise ...
7
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0answers
152 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 ...
6
votes
4answers
879 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 ...
6
votes
2answers
277 views

Who can prove that a triangular number cannot be a cube, fourth power or fifth power?

Triangular numbers (See https://en.wikipedia.org/wiki/Triangular_number ) are numbers of the form $$\frac{n(n+1)}{2}$$ In ProofWiki I found three claims about triangular numbers. The three claims ...
6
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2answers
122 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-...
6
votes
1answer
221 views

Small powers of small numbers: Is there really a pattern or am I just used to looking for them?

This probably is just a coincidence, but I've always found it interesting and I wanted to put some feelers out there to see if maybe there really is something to it after all. There are these ...
6
votes
2answers
269 views

Can $10^n+89$ ever be a Perfect Square for $n>3$?

Is it correct that if an natural number $10^n+89$ is a perfect square then $n=3$? The answer is clear if $n$ is an even number. For odd $n$ I can prove that $10^n+89$ can be a perfect square only if $...
6
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0answers
95 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 ...
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 ...
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+...
6
votes
0answers
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 ?
6
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0answers
241 views

Prime Powers and Differences of Consecutive Cubes

I am wondering if it has been proven that there does not exist a prime $p$ and an integer $r \ge 3$ such that $p^r = (n + 1)^3 - n^3$ for some integer $n$. Note that this is a special case of Beal's ...
5
votes
4answers
214 views

Prove that $\sqrt {2^n-1}$ is irrational for every integer $ n>1$

Prove that $\sqrt {2^n-1}$ is irrational for every integer $ n>1$ I tried assuming it was equal to $\frac p q $. I get $2^nq^2-q^2 = p^2 $ But I don't see where to go from there.
5
votes
4answers
129 views

Prove by contradiction that $(n+1)^3 \not= n^3 +(n-1)^3$ for $3$ consecutive positive integers

Prove by contradiction that if $n-1$, $n$, $n+1$ are consecutive positive integers, then the cube of the largest cannot be equal to the sum of the cubes of the other two. Assume that: $$ (n+1)^3 = (n-...
5
votes
2answers
289 views

proof - Show that $1! +2! +3!+\cdots+n!$ is a perfect power if and only if $n=3$

Show that $1! +2! +3!+\cdots+n!$ is a perfect power if and only if $n =3$ For $n=3$, $1!+2!+3!=9=3^2$. I also feel that the word 'power' makes it a whole lot hard to prove. How do we prove this? What ...
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 ...
5
votes
2answers
105 views

Proving the irrationality of the concatenation of the $n$th powers of primes

Note: The apostrophes are meant to separate different groups of digits. Like, $0.{1^2}'{2^2}'{3^2}'{4^2}'\cdots=0.14916\cdots$. I wasn't able to come up with something better. It is easy to show $0.{...
5
votes
2answers
202 views

Finding a prime number $p$ and $x, y, z\in \mathbb N$ such that $x^p+y^p=p^z$

I'm interested in Fermat's Last Theorem (Wiles theorem?). Then, I made the following similar question: Find a prime number $p$ and natural numbers $x, y, z$ such that $x^p+y^p=p^z$. I got that the ...
5
votes
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 ...
5
votes
1answer
83 views

What values can $2^j-3^k$ have?

What values can $2^j-3^k$ have? E.g., $$ 2^2-3^1=1\\ 2^2-3^0=3\\ 2^3-3^1=5\\ 2^4-3^2=7 $$ Can every number not divisible by $2$ or $3$ be written as $2^j-3^k$? If not, why?
5
votes
2answers
128 views

$s(n^x)$ is not a perfect square for all $x$

Does there exist an $n$ such that $s(n^x)$ is not a perfect square for all positive integers $x$ where $s(m)$ denotes the sum of the digits of a positive integer $m$? If $n = 5$, for example, then ...