Lucas's theorem

Builds on Congruence modulo n Prime number

Statement

Let $p$ be a prime number, and let $m$ and $n$ be natural numbers with base- $p$ digits $m = \sum_{i=0}^{k} m_i p^i$ and $n = \sum_{i=0}^{k} n_i p^i$ , where $0 \le m_i, n_i < p$ . Then $\binom{m}{n} \equiv \prod_{i=0}^{k} \binom{m_i}{n_i} \pmod p,$ with the convention $\binom{a}{b} = 0$ when $b > a$ .

Remarks

A consequence: $p \mid \binom{m}{n}$ if and only if $n_i > m_i$ for some digit $i$ . For example, with $p = 3$ : $10 = 101_3$ and $4 = 011_3$ . The theorem gives $\binom{10}{4} \equiv \binom{1}{0} \binom{0}{1} \binom{1}{1} = 0 \pmod 3$ , and indeed $\binom{10}{4} = 210$ . For $p = 2$ , the binomial coefficient $\binom{m}{n}$ is odd if and only if $m$ has a binary digit $1$ at each place where $n$ has one.

History

Édouard Lucas published the theorem in 1878.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Divisibility definition
Congruence modulo n definition
Prime number definition
Lucas's theorem statement

Tag	`0RY4GGMEQG`
URI	`m2://0RY4GGMEQG`
Kind	statement, theorem
Writing	W1 stated
MSC	11B65
Status	This copy of the site has no build data. Run `m2 status 0RY4GGMEQG`.
Uses	Congruence modulo n, Prime number
Same as	Wikidata Q1186808 Wikipedia Lucas's theorem

Lucas's theorem

Statement

Remarks

History

Reading path

Further reading

What this page needs for W2

What links here