On 7/16/07, David Wilson <davidwwilson@comcast.net> wrote:
It looks as if B(p) == 2 (mod p) for prime p.
Is this true, and are there "Bell pseudoprimes" in this respect?
This is the special case n = 0 of the "Touchard recurrence" B(n+p) = B(n+1) + B(n) mod p for all natural n and prime p. It is easily proved from the "umbral" identity B^n = (B+1)^n usually used to define the Bell numbers.
From this incidentally it follows that they are periodic modulo a prime: B(n + (p^p-1)/(p-1)) = B(n) mod p; it is conjectured that the above equation gives the minimum period.
This has been verified for p <= 101 by by S.S.Wagstaff --- I don't have the reference, but it should be easy to locate on the web. There is an earlier paper in Math. Comp. (circa 1987) explaining the background, which I can dig out for anybody who's interested. Fred Lunnon