The question is in the title. It was asked by Paul Erdős, e.g. as part of Section 9 in this [paper](https://www.ime.usp.br/%7Eyoshi/resenhas/abstracts/Erdos.pdf).



Does the equation $\varphi(n)=\sigma(m)$ have infinitely many solutions?
Yes, this was proved by [Ford, Luca and Pomerance](https://faculty.math.illinois.edu/%7Eford/wwwpapers/phi=sig.pdf) in 2010 (paper in Bulletin of the London Math. Soc.).


