Let $S=\{x_1,\cdots ,x_n\}$ be a set of $n$ distinct positive integers and $e\ge 1$ an integer. Denote the $n\times n$ power GCD (resp. power LCM) matrix on $S$ having the $e$-th power of the greatest common divisor $(x_i,x_j)$ (resp. the $e$-th power of the least common multiple $[x_i,x_j]$) as the $(i,j)$-entry of the matrix by $\left({(x_i,x_j)}^e\right)$ (resp. $\left({[x_i,x_j]}^e\right))$. We call the set $S$ an odd gcd closed (resp. odd lcm closed) set if every element in $S$ is an odd number and $(x_i,x_j)\in S$ (resp. $[x_i,x_j]\in S$) for all $1\le i,j\le n$. In studying the divisibility of the power LCM and power GCD matrices, Hong conjectured in 2004 that...

In this paper we investigate the solutions of the equation in the title, where $\phi$ is the Euler function. We first show that it suffices to find the solutions of the above equation when $m=4$ and $x$ and $y$ are coprime positive integers. For this last equation, we show that aside from a few small solutions, all the others are in a one-to-one correspondence with the Fermat primes.

For a positive integer $n$ we write $\phi \left(n\right)$ for the Euler function of $n$. In this note, we show that if $b>1$ is a fixed positive integer, then the equation $$\phi \left(x\frac{b^n-1}{b-1}\right)=y\frac{b^m-1}{b-1},\text{where}x,y\in \{1,...,b-1\},$$ has only finitely many positive integer solutions $(x,y,m,n)$.

For a prime p > 2, an integer a with gcd(a,p) = 1 and real 1 ≤ X,Y < p, we consider the set of points on the modular hyperbola ${}_{a,p}(X,Y)=(x,y):xy\equiv a\left(modp\right),1\le x\le X,1\le y\le Y$. We give asymptotic formulas for the average values ${\sum }_{\begin{array}{c}(\end{array}x,y){\in }_{a,p}(X,Y)x\ne y*}\phi \left(\right|x-y\left|\right)/|x-y|$ and ${\sum }_{\begin{array}{c}(\end{array}x,y){\in }_{a,p}(X,X)x\ne y*}\phi \left(\right|x-y\left|\right)$ with the Euler function φ(k) on the differences between the components of points of ${}_{a,p}(X,Y)$.