Math 2.0

Number theory Congruences Theorem

Wilson's theorem

Builds on Prime numberCongruence modulo n

Statement

Let pp be a prime number. Then (p−1)!≡−1(modp)(p - 1)! \equiv -1 \pmod p.

Remarks

A proof: in ℤ/pℤ\mathbb{Z}/p\mathbb{Z}, each element of {1‾,…,p−1‾}\{ \overline{1}, \dots, \overline{p - 1} \} has an inverse. Only 1‾\overline{1} and −1‾\overline{-1} are their own inverses. The other elements cancel in pairs in the product. For example, 4!=24≡−1(mod5)4! = 24 \equiv -1 \pmod 5 and 6!=720≡−1(mod7)6! = 720 \equiv -1 \pmod 7. The converse of Wilson's theorem also holds, so the congruence characterizes the primes.

History

Ibn al-Haytham stated the result about the year 1000. Edward Waring announced it in 1770 and named his student John Wilson as its discoverer. Lagrange gave the first proof in 1771.

Reading path

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

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Divisibility, definition, no evidence levelDivisibilitydefinitionCongruence modulo n, definition, no evidence levelCongruence modulo ndefinitionPrime number, definition, no evidence levelPrime numberdefinitionWilson's theorem, statement, no evidence levelWilson's theoremstatement
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  • no evidence level
  1. Divisibility definition
  2. Congruence modulo n definition
  3. Prime number definition
  4. Wilson's theorem statement

Where this leads

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Further reading

  1. Web https://en.wikipedia.org/wiki/Wilson%27s_theorem, lead section
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Tag03KGGKTB2E
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Kindstatement, theorem
WritingW1 stated
MSC11A07
StatusThis copy of the site has no build data. Run m2 status 03KGGKTB2E.
UsesPrime number, Congruence modulo n
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