Fungrim entry: a4d6fc

B_{n + a} \equiv B_{n} \pmod {m}\; \text{ where } a = \begin{cases} 3, & m = 2\\13, & m = 3\\12, & m = 4\\781, & m = 5\\39, & m = 6\\ \end{cases}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \{2, 3, \ldots, 6\}

TeX:

B_{n + a} \equiv B_{n} \pmod {m}\; \text{ where } a = \begin{cases} 3, & m = 2\\13, & m = 3\\12, & m = 4\\781, & m = 5\\39, & m = 6\\ \end{cases}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \{2, 3, \ldots, 6\}

Definitions:

Fungrim symbol	Notation	Short description
`BellNumber`	$B_{n}$	Bell number
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints

Source code for this entry:

Entry(ID("a4d6fc"),
    Formula(Where(CongruentMod(BellNumber(Add(n, a)), BellNumber(n), m), Equal(a, Cases(Tuple(3, Equal(m, 2)), Tuple(13, Equal(m, 3)), Tuple(12, Equal(m, 4)), Tuple(781, Equal(m, 5)), Tuple(39, Equal(m, 6)))))),
    Variables(n, m),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(m, Range(2, 6)))))

Topics using this entry

Bell numbers