Problem 479

Erdős Problem #479: declared status 'open'. Formalized: yes. Is it true that, for all $k\neq 1$, there are infinitely many $n$ such that $2^n\equiv k\pmod{n}$? Current best: Erd\H{o}s and Graham report that Graham, Lehmer, and Lehmer have proved this for $k=2^i$ for $i\geq 1$, or if $k=-1$, but I cannot find such a paper. Prize: no. OEIS: A006517, A006521, A015919, A015921, A015940, A036236, A050259. Tags: number theory.