The following proposition provides also a very inefficient method of prooving if an integer is a prime number.

**Proposition**: A Necessary and Sufficient Condition for an Integer to be Prime

An integer $n > 1$ is a prime number if and only if the following congruence holds:

$$(n-1)!\equiv -1\mod n.$$

### Notes

- In this equation, $(n-1)!$ denotes the factorial of $(n-1).$
- This proposition is also known as
**Wilson’s theorem**, called after its discoverer John Wilson (1741 – 1793).

| | | | | created: 2019-05-11 19:54:19 | modified: 2019-05-12 09:17:37 | by: *bookofproofs* | references: [1272]

## 1.**Proof**: *(related to "A Necessary and Sufficient Condition for an Integer to be Prime")*

(none)

[1272] **Landau, Edmund**: “Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie”, S. Hirzel, Leipzig, 1927

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