a) Find all solutions of the equation $p^2+q^2+r^2=pqr$, where $p,q,r$ are positive primes. b) Show that for every positive integer $N$, there exist three integers $a,b,c\geq N$ with $a^2+b^2+c^2=abc$.
Source: Dürer Competition Finals 2023/E+ 2
Tags: number theory, Diophantine equation
a) Find all solutions of the equation $p^2+q^2+r^2=pqr$, where $p,q,r$ are positive primes. b) Show that for every positive integer $N$, there exist three integers $a,b,c\geq N$ with $a^2+b^2+c^2=abc$.