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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Why is it that $(n+(n+1)^2+(n+2)^3) \bmod 144 =0$ starting with $n=3$ and incrementing $n=n+6$?

The series involves: $(n+(n+1)^2+(n+2)^3) \bmod 144 =0$ starting with $n=3$ and incrementing $n=n+6$ Examples: $3+4^2+5^3 = 3+16+125 =144$ and $144 / 144 =1$ $9+10^2+11^3 = 9+100+1331 =1440$ and $1440 ...
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Why is the following for $B= A^x -(A-1)^x-(A-2)^x-(A-3)^x…(A-n)^x$ as long as $B ≥ 0$?

As a web developer that programs in PHP, on my free time I like to run some math algorithms to see the wonder of math. In this experiment, I wanted to check the final results for $B= A^x -(A-1)^x-(A-2)...
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300 views

Solutions to $A^N+B^N=C^N \pm 1$

Is there a solution to $A^N+B^N=C^N \pm 1$ where $A,B,C,N\in\Bbb{N}$, such that $A,B,C > 1,N \geq 4$ and $gcd(A,B)=gcd(B,C)=gcd(A,C)=1$? This question was inspired by this one: $A^X+B^Y=C^Z\pm 1$ ...
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Power Tower and Finite Tetration

Let $$\underbrace{a^{a^{.^{.^{.}}}}}_{2020} =2020$$ How can I solve for $a$? Had this been an infinite power tower, I could have used the self-replication. But the finite number of terms (i.e. $2020$ ...
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When do we have $(x y)^2 = x^2 y^2 $?

I just started thinking about algebra so this might be a trivial question. Anyway, Under what conditions do we have $$(x y)^2 = x^2 y^2 $$ ? Does it need to be a group ? Or a groupoid ? Or a monoid ? ...
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Are there more perfect powers of this form?

Let $a,b$ be integers , $a\ge 3$ , $b\ge 1$ If $f_a$ is the $a$ th cyclotomic polynomial, are there any perfect powers of the form $f_a(b)$ apart from $121$ and $343$ ? In the range $$3\le a\le 1\ ...
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Comparing powers of $2$ and $5$

Is it possible to prove that $5^{152}<2^{353}$ and $2^{1413}<3\cdot 5^{608}$ without using a calculator or logarithms (middle school math only recommended)? My idea for the first one was to use ...
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Finding remainder using Fermat's Little Theorm

Use Fermat's Little Theorem to find the remainder of $5^{15}$ divided by $1337$. I know Fermat's Little Theorem, but failed to understand how to use it in this. output.944
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Two questions about a special function

Define $$f(n):=\sum_{j=1}^n j^{n+1-j}=1+2^{n-1}+3^{n-2}+\cdots+(n-1)^2+n$$ for a positive integer $n$. With PARI/GP, this function can be calculated with the self-defined function ...
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$m^n$ has exactly $9$ digits, all of these digits are distinct.

I was reading the following problem in a book: $2^{29}$ has exactly $9$ digits, all of these digits are distinct, what is the missing digit? I tried to solve it, and I did it correctly. Suddenly, the ...
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Solving $\tau^4 +1=u^2 +v^2$ over the integers

I'm trying to show, that the solutions $t_i$ from Pell's equation $t^2-2s^2=\pm 1$ are not squares (only for the trivial cases). For the even solutions $t_{2i}$ (solutions for the + sign) this is easy,...
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Perfect powers in Horadam sequence

The Horadam sequence $\{W_n\}$ is defined by the Binet formula [Using notation from here]: $$W_n=\frac{A\alpha^n-B\beta^n}{\alpha-\beta}$$ where, $$A\ =\ b\ -\ a\beta \text{ and } B\ =\ b\ -a\alpha$$ $...
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$\lfloor x^n\rfloor\lfloor y^n\rfloor$ is a perfect square

Let $x,y\ge 1$ be non-integer real numbers such that $\lfloor x^n\rfloor\lfloor y^n\rfloor$ is a perfect square for all natural numbers $n$. Does it follow that $x=y$? From this question we know the ...
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Has this family of Diophantine equations ever been studied?

This might be a trivial question, so, if it is, I will delete it as soon as I get an answer. I try to come up with conjectures once in a while. After all, the sleep of reason produces monsters (...
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Infinitely many consecutive powers with no consecutive equal digits?

Let $P_k$ be the set of all $n\in\mathbb N$ such that $n^1,n^2,\dots,n^k$ have no consecutive equal digits. In other words, their decimal expansion does not contain '$00$', '$11$' ,'$22$', $\dots$, '$...
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Extending Given Digits to make Perfect Powers

I read this article titled Extending Given Digits to make Primes or Perfect Powers by Sury B, which appeared in the Resonance periodical (October 2010, Indian Statistical Institute, Bangalore). In ...
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Power values of polynomial

$f(x) = a_0 + a_1 x + a_2 x^2 + \dots + a_n x^n$ is a polynomial of degree $n$ with positive integer coefficients. Primary problem statement: Is the Exponential Diophantine Equation $f(f(a) + 1) = y^...
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How to check if a number can be represented as the sum of two consecutive perfect cubes.

How to check if a number can be represented as the sum of two consecutive perfect cubes. for eg N = 35 can be represented as the sum of two consecutive perfect cubes 23 and 33
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If there is a way to express an integer as a sum of three non-zero cubes, then there are infinitely many ways

How to prove the following statement: "If an integer can be expressed as a sum of three non-zero cubes in a way, then it can be expressed as a sum of three non-zero cubes in infinitely many ways.&...
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Can a multiperfect number be a perfect power?

(Note: The following post is an offshoot of this earlier MSE question: Can a multiperfect number be a perfect square?.) Let $\sigma(x)$ denote the classical sum of divisors of the positive integer $x$...
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Find all positive integers $n$ for which $1372n^4 - 3 $ is an odd perfect square.

Find all positive integers $n$ for which $1372\,n^4 - 3$ is an odd perfect square. I tried $\bmod ,4,5,7$ and failed. Next, I used Vieta’s Theorem and failed again. Any hints, please. Thank you very ...
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Find all $n$ which $7(n^2 + n + 1)$ is perfect $4^{th}$ power.

Find all positive integer $n$ , which $7(n^2 + n + 1)$ is perfect $4^{th}$ power. What I tried Let $7(n^2 + n + 1) = a^4$ $\to$ $ 7 | a$ and $a$ is odd. We then get $(n^2 + n + 1) = 343k^4$ ; $k \in \...
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Can you find a function $\beta(x)$ where if $a+b=n^m$ then $\beta(\frac{a}{b})$ is irrational?

Can you find a function $\beta(x)$ where if $a+b=n^m$ then $\beta(\frac{a}{b})$ is irrational but if $a+b$ isn't equal to $n^m$ then it is rational ($a$ and $b$ are co-prime)? $n>0$ and $m>1$ $m$...
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Finding an infinite sequence of natural numbers for which any finite partial sum avoids being a perfect power.

$\textbf{Question:}$Is there an infinite set of positive integers such that no matter how we choose some elements of this set, their sum is not a perfect power? $\textbf{My progress:}$I thought of ...
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Techniques for solving an equation $f(x)=y^n$ over the integers?

Despite of the negative result of Hilbert's tenth problem, diophantine equations of the form $$f(x)=y^n$$ can often be fully solved. Here, $f(x)$ is a polynomial with integer coefficients and $n>1$ ...
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Perfect powers nearest to factorials

Suppose, $n\ge 8$ is an integer. Let $s$ be the smallest non-negative integer such that $n!-s$ is a perfect power. Let $t$ be the smallest non-negative integer such that $n!+t$ is a perfect power. ...
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Is there another perfect power?

This is related to this question : Can we conclude $n=p-1$? For which positive integers $\ n\ $ is $$(2n)!+n!+1$$ a perfect power ? For $\ n=1\ $ and $\ n = 2\ $ , we have a perfect power and there is ...
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Closest sums of pairs of reciprocals of prime numbers

This question is related to an upper bound for a problem from my five-year-old question. Let $n\ge 3$ be a natural number and $A_n$ be a set consisting of $1$ and first $n-1$ prime numbers. We want to ...
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How can we prove that $3t$ cannot be a perfect cube for any integer $t$ except 9?

If $t \in \mathbb{Z}$ then prove that $3t$ can never be a perfect cube except for $t=9$. How can we prove things like these? I’m pretty new to Number-Theory and I find it difficult to prove things ...
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There exist $n$ different integers in the interval $\big(k^n,(k+1)^n\big)$ whose product is a perfect $n$-th power.

Given a positive integer $ n> 2 $. Prove that there exists a natural number $ K $ such that for all integers $ k \ge K $ on the open interval $ \big({{k} ^{n}}, \ {{(k + 1)} ^{n}}\big) $ there are $...
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How to find the sum of the remainders of a number modulo all the perfect powers less than it?

Perfect powers are numbers of the form $ x^y $ where $x\geq1$ and $y>1$. For example, perfect powers are $\{1, 4, 8, 9, 16, 25, 27, 32, \cdots \}$. If $n$ is a natural number, then we denote by $P(...
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Sum of perfect powers of n natural numbers

The perfect power of an integer $n = p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_r^{\alpha_r}$ are $p_1^2, p_1^3\cdots p_1^{\alpha_1},p_2^2, p_2^3 \cdots p_2^{\alpha_2},(p_1 p_2)^2, (p_2 p_3)^2\cdots$ ...
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For any $k \gt 3$, if $n!+k$ is a perfect power then does there exist any $n\gt k$?

A while ago, I asked a similar question For any $k \gt 1$, if $n!+k$ is a square then will $n \le k$ always be true? where users mathworker21 and WE Tutorial School proved that for non-square $k$, $n\...
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Surprising fact about a certain number-theoretic function

Ante suggested the following function : For natural number $n$ we can observe the $n$ remainders $b_1,...,b_n$ by writing $n$ as $n=a_k \cdot k+b_k$ for $1 \leq k \leq n$ Because of the familiar ...
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what are the different ways to check if a number is perfect cube

What are the different ways to check if a number is a perfect cube? I was thinking some ways but want to know different ways to check if a number is a perfect cube or not? And is there any way to ...
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Find all positive integers satisfying $a^{b^2}=b^a$.

So the question is : Determine all pairs $(a, b)$ of positive integers satisfying $a^{b^2} = b^a$. I tried it for 2 hours by different methods like taking it even, odd and by modulus method but ...
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How to find power of a given number?

65536= 4^8------( how can i find the base and power for a number.) Now, if I am randomly given a number. Say, 4096 and I need to find a common base having some power. I mean a^n. Where a is the base ...
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About primes and their powers in bases $\{2,3,4,5,6,7,8,9,10\}$

For some prime $p=p_{10}$, where $p_{10}$ means just that that prime is represented in base $10$, if: *) In at least one base from the set $\{2,3,4,5,6,7,8,9\}$ the number $p^2$ is prime (carefully ...
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Is there for all $N$ an integer that is the sum of, say, twenty $N$-th powers in two different ways?

I believe the following to be true, but I have been unable to find any reference concerning the question. Statement : for every $k\geq 1$, there exists $N = N(k)$ such that there is no non-trivial ...
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$f(x)= \frac{1}{x^{2}-3x-4} $The power series expansion at x = 1.

$f(x)= \frac{1}{x^{2}-3x-4} $ The power series expansion at x = 1. I made an answer, but I am not sure if it is right. $$1/5\left[\sum_{n=0}^∞ (-1)^{(n+1)} -\sum_{n=0}^∞ 1/4^{(n+1)}\right]$$ This is ...
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Odd power of 2 plus even power of 2, is divisible by 6

Prove $2^{2m+1}+2^{2n}$ is divisible by 6 for all integers $n\geq1$ and $m\geq0$. Equivalently one can prove $2+4^k$ is divisible by 3.
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Can we reduce the diophantine equation $x^4+4y^4=z^2$ to $x^4+y^4=z^2$ to show there is no nontrivial solution?

Is it possible to reduce the diophantine equation $$x^4+4y^4=z^2,$$ to the diophantine equation $$x^4+y^4=z^2,$$ to show that there is no nontrivial solution? Please assume the second equation is ...
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Solving Diophantine Equations Involving Exponentials as Positive Integers

Find all positive integers $x$, $y$ and $z$ such that $$x^5+4^y=2013^z$$ Now as $$11|2013 \implies 11|4^y+x^5$$ Now by Euclid's Division Lemma we have $y=5q, 5q+1, 5q+2, 5q+3, 5q+4$, so we get $$4^y \...
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What is the $n^\text{th}$ perfect power, $P(n)$?

Let $P(n)$ denotes the $n^\text{th}$ perfect power of natural numbers (in ascending order without repetition). So, $P(1)=1, P(2)=4, P(3)=8, P(4)=9, P(5)=16, P(6)=25, P(7)=27, P(8)=32, \dots$. Is ...
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Conjecture That The Only Nearly Perfect Numbers Are The Powers Of 2

Call a natural number $n$ “flawed” if its aliquot sum (the sum of its factors less than itself) is $n-1$. Show that the set of flawed numbers is equivalent to the set of perfect powers of $2$. ...
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Pythagorean triple with hypotenuse a power of $2$

Is there a Pythagorean triple whose largest element is a power of 2? That is: are there solutions to $a^2 + b^2 = 2^{2k}$ in the positive integers?
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Find all natural number $n$ for which $3^9+3^{12}+3^{15}+3^n$ is a perfect cube.

Find all natural number $n$ for which $3^9+3^{12}+3^{15}+3^n$ is a perfect cube. What I have tried. $3^9+3^{12}+3^{15}+3^n$ $=3^9(757+3^{n-9})$ Let $757+3^{n-9}=a^3$ Taking modulo $3$: $a^3\...
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Are there forced factors of numbers of this kind?

Let $$f(n):=n^{n^2}+(n+1)^{(n+1)^2}$$ for a positive integer $n$. For clarification $n^{n^2}$ means $n^{(n^2)}$ , analogue for the other summand. Can we find a concrete factor of $\ f(n)\ $ (like ...
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Can $n$ be expressed as $a^b-c^d$, where $a,b,c,d,$ and $n$ are natural numbers, not necessary distinct?

Can $n$ be expressed as $a^b-c^d$, where $a,b,c,d,$ and $n$ are natural numbers, not necessary distinct, and $b$ and $d$ can not be both equal to $1$? For example, when $n=1319$, then $2^{11}-3^6=...
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How many zeros in the decimal representation of $5^n$?

I'm curious about some properties of the powers of 5 $$5^2=25,\quad5^3=125,\quad 5^4=625,\quad 5^5=3125,\quad ...$$ Is it true that at least $50$% of the digits in the decimal representation of $5^n$ ...

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