On an iterated arithmetic function problem of Erdos and Graham

Abstract

Erd\H{o}s and Graham define $g(n) = n + \phi(n)$ and the iterated application $g_k(n) = g(g_{k-1}(n))$. They ask for solutions of $g_{k+r}(n) = 2 g_{k}(n)$ and observe $g_{k+2}(10) = 2 g_{k}(10)$ and $g_{k+2}(94) = 2 g_{k}(94)$. We show that understanding the case $r = 2$ is equivalent to understanding all solutions of the equation $\phi(n) + \phi(n + \phi(n)) = n$ and find the explicit solutions $ n = 2^{\ell} \cdot \left\{1,3,5,7,35,47\right\}$. This list of solutions is possibly complete: any other solution derives from a number $n=2^{\ell} p$ where $p \geq 10^{10}$ is a prime satisfying $\phi((3p-1)/4) = (p+1)/2$. Primes with this property seem to be very rare and maybe no such prime exists.