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 $$(2,2,1,1)$$

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?

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, […]

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? 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.

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, […]

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

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 […]

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, […]

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 […]

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