Articles of perfect powers

Equation over the integers

Find all quartuples $(a,b,c,d)$ with non-negative integers $a,b,c,d$ satisfying $$2^a3^b-5^c7^d=1$$ One solution is $$(2,2,1,1)$$

The smallest integer whose digit sum is larger than that of its cube?

79 is an example of a number whose digital sum is greater than that of its square (6241). Which is the least number, if any, whose digital sum is greater than that of its cube?

Prime candidates of the form $n^{(n^n)}+n^n+1$?

Let $$\large f(n)=n^{(n^n)}+n^n+1$$ Checking $f(n)$ for $2\le n\le 100$, I noticed that $f(n)$ has a small prime factor except for $n=12,53$ and $60$ For $n=53$, I found the prime factor $7074407$ , but for $n=12$ and $n=60$, I did not find a prime factor yet. The numbers are too large to apply a primilaty test, […]

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 $${1 \above 1.5 pt n^2}\sum_{i=1}^n \sum_{j=1}^n a_{ij} \leq {1 \above 1.5pt 3}$$ with equality holding if and only if $n=3$ or $n=6$ ? Let $A(n)$ be a […]

What better way to check if a number is a perfect power?

What better way to check if a number is a perfect power? Need to write an algorithm to check if $ n = a^b $ to $ b > 1 $. There is a mathematical formula or function to calculate this? I do not know a or b, i know only n.

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 followings are sufficient : for any non-negative integer $d$,$$(p,x,y,z)=(3,3^d,2\cdot3^d,2+3d),(3,2\cdot3^d,3^d,2+3d),(2,2^d,2^d,1+2d).$$ However, I’m not sure that these are the only solutions I want. Then, […]

Solving a little Diophantine equation:$(n-1)!+1=n^m$

This question already has an answer here: To find all $n$ such that $(n-1)!+1$ is a perfect power of $n$ 1 answer

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

Possible Duplicate: Number theory proving question? Dear friends, Since $b$, $x$, $y$, $z$, $\ldots$ are integers greater than 1, how can we prove that $$ \gcd (b ^ x – 1, b ^ y – 1, b ^ z – 1 ,\ldots)= b ^ {\gcd (x, y, z, .. .)} – 1 $$ ? Thank […]

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 $$r=\frac{a^\color{red}{3}+b^\color{red}{3}}{c^\color{red}{3}+d^\color{red}{3}}.$$ For $r=p/q$ where $p,q$ are positive integers, we can take $$(a,b,c,d)=(3ps^3t+9qt^4,\ 3ps^3t-9qt^4,\ 9qst^3+ps^4,\ 9qst^3-ps^4)$$ where $s,t$ are positive integers such that $3\lt r\cdot(s/t)^3\lt 9$. For $r=2014/89$, for example, […]

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. $$8,1,15,10,6,3,13,12,4,5,11,14,2,7,9$$ Also, a few days ago, a friend of mine taught me that one can arrange all the numbers from $1$ to $\color{red}{305}$ in a row […]