OFFSET
0,1
COMMENTS
Jacques Touchard proved in 1933 that for the Bell numbers (A000110), Bell(p+k) == Bell(k+1) + Bell(k) (mod p) for all primes p and k >= 0.
a(29) > 242000 and a(89) > 90000, if they exist. The terms from a(30) to a(89) are 2, 163, 2, 2, 2, 7, 2, 19, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 7, 2, 2, 2, 359, 2, 3, 2, 2, 2, 7, 2, 43, 2, 2, 2, 3, 2, 5, 2, 2, 2, 5, 2, 547, 2, 2, 2, 3, 2, 7, 2, 2, 2, 59, 2, 5, 2, 2, 2. - Giovanni Resta, Aug 26 2018
a(29) > 10^7. - Hiroaki Yamanouchi, Sep 01 2018
REFERENCES
J. Touchard, "Propriétés arithmétiques de certains nombres récurrents", Ann. Soc. Sci. Bruxelles A 53 (1933), pp. 21-31.
LINKS
Eric Weisstein's World of Mathematics, Touchard's Congruence
EXAMPLE
The smallest prime p such that Bell(p+1) == Bell(2)+Bell(1)(mod p^2) is 5, since Bell(6) - Bell(2) - Bell(1) = 203 - 2 - 1 = 200 = 5^2 * 8, thus a(1) = 5.
MATHEMATICA
a = {}; n = 0; While[n < 101, p = 2; While[!Divisible[BellB[p + n] - BellB[n] - BellB[n + 1], p^2], p = NextPrime@p]; a = AppendTo[a, p]; n++]; a
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Amiram Eldar, May 12 2017
STATUS
approved