The infamous Collatz conjecture. For a detailed discussion of the history and theory surrounding this problem we refer to the overview by Lagarias \cite{La10}.

This is not a problem due to Erd\H{o}s; it was first devised by Collatz before 1952. Erd\H{o}s referred to this problem on several occasions as 'hopeless'. As Lagarias \cite{La16} notes, the closest Erd\H{o}s ever came to working on problems of this nature is the theorem described in the remarks to [1134].

It is often claimed that Erd\H{o}s offered \$500 for a solution to this problem; this claim originated in a survey article by Lagarias \cite{La85}.

Lagarias reported, in personal communication, that this came from a conversation he had with Erd\H{o}s and Graham around 1983, in which Graham asked Erd\H{o}s to make an estimate of what value Erd\H{o}s would put the problem on his prize scale, to which Erd\H{o}s replied \$500. Therefore, strictly speaking, Erd\H{o}s never offered \$500 specifically as a prize, but we include this prize value here for comparing those problems which Erd\H{o}s rated as 'prize problems'.

This is Problem E16 in Guy's collection \cite{Gu04}, in which Guy quotes Erd\H{o}s as saying "Mathematics may not be ready for such problems".