Everything about Bell Number totally explained
In
combinatorial mathematics, the
nth
Bell number, named in honor of
Eric Temple Bell, is the number of
partitions of a
set with
n members, or equivalently, the number of
equivalence relations on it. Starting with
B0 =
B1 = 1, the first few Bell numbers are :
» 1,
1,
2,
5,
15,
52, 203, 877, 4140, 21147, 115975
(See also
breakdown by number of subsets/equivalence classes.)
Partitions of a set
In general,
Bn is the number of
partitions of a set of size
n. A partition of a set
S is defined as a set of nonempty, pairwise disjoint subsets of
S whose union is
S. For example,
B3 = 5 because the 3-element set such that 3 isn't together in one class with element 2: for counting partitions two elements which are always in one class can be treated as just one element. The 3 appears in the previous row of the table.
Prime Bell numbers
The first few Bell numbers that are primes are:
» 2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837
corresponding to the indices 2, 3, 7, 13, 42 and 55
The next prime is
B2841, which is approximately 9.30740105 × 10
6538.
(External Link
) As of 2006, it's the largest known prime Bell number. Phil Carmody showed it was a
probable prime in 2002. After 17 months of computation with Marcel Martin's
ECPP program Primo, Ignacio Larrosa Cañestro proved it to be prime in 2004. He ruled out any other possible primes below
B6000, later extended to
B30447 by Eric Weisstein.
Further Information
Get more info on 'Bell Number'.
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