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Fungrim entry: b41c49

Bn+pmmBn+Bn+1(modp)B_{n + {p}^{m}} \equiv m B_{n} + B_{n + 1} \pmod {p}
Assumptions:nZ0  and  pP  and  mZ1n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; p \in \mathbb{P} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 1}
B_{n + {p}^{m}} \equiv m B_{n} + B_{n + 1} \pmod {p}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; p \in \mathbb{P} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 1}
Fungrim symbol Notation Short description
BellNumberBnB_{n} Bell number
Powab{a}^{b} Power
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
PPP\mathbb{P} Prime numbers
Source code for this entry:
    Formula(CongruentMod(BellNumber(Add(n, Pow(p, m))), Add(Mul(m, BellNumber(n)), BellNumber(Add(n, 1))), p)),
    Variables(n, p, m),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(p, PP), Element(m, ZZGreaterEqual(1)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC