Wilson’s theorem (number theory) . In mathematics, especially in number theory, there is a proposition that links three concepts: primality, factorial of a nonzero integer and congruence of numbers with respect to a module. This is Wilson’s theorem .
Summary
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- 1 Statement
- 2 Test
- 1 Example
- 3 Sources
Statement
(m-1)! ~ -1 (mod m) as long as m is rational prime.
- Indeed the reciprocal of this theorem can be easily demonstrated, in implicative language, from
(k-1)! ~ -1 (mod k) it follows that k is prime, if that did not happen k and (k-1)! they have a common divisor, which is a contradiction. Therefore, the theorem provides a necessary and sufficient condition for k to be a prime number.
Proof
Let’s consider the set of numbers
1, 2, 3, …, m-2, m-1 and let the pair of 2 be the number b2 in the list for which 2 a2 ~ 1 (mod m) we know that such a number exists since 2 is a prime relative to m. Then we can link 3 with your partner and so on. If h were its own partner, h squared would be congruent to 1 mod m, where h 2 -1 = 0, a fact that requires h = m-1 or
h = 1. Then discarding these two numbers we have:
(m-2)! ~ (2 b2) () ()… () (mod m),
Where in each parenthesis there are two numbers whose product is congruent to 1 mod m, from where:
(m-1)! ~ 1 (1) (1)… (1) (p-1) ~ -1 (mod)
Example
We can clarify the above with an example. Let m = 11. This implies 2y ~ 1 (mod 11) has a solution = 6, so 2 and 6 constitute a pair, the pair of 4 is 3 and others. It should be written:
10! = 1 (2.6) (3.4) (5.9) (7.8) (10) = 1.1.1.1.1. (- 1) ~ -1.
