Is there a Bell number $B\_n$ of the form $2^k$ for some $k>1$? If there is, are there infinitely many?



Can a Bell number be a power of 2?
No. It's easy to see that $B\_n=4$ is impossible, and $B\_n$ is never divisible by 8. This follows from the fact that $B\_n$ is periodic modulo 8 with period 24. See, e.g., W. F. Lunnon, P. A. B. Pleasants, and N. M. Stephens, [Arithmetic properties of Bell numbers to a composite modulus I](http://matwbn.icm.edu.pl/ksiazki/aa/aa35/aa3511.pdf/), Acta Arithmetica 35 (1979),
1–16, Theorem 6.4.


The Bell numbers $B\_n \bmod 8$ for $n$ from 0 to 23 are
1, 1, 2, 5, 7, 4, 3, 5, 4, 3, 7, 2, 5, 5, 2, 1, 3, 4, 7, 1, 4, 7, 3, 2. So $B\_n$ is never 0 or 6 modulo 8.


