# Wilson's Theorem

Wilson's Theorem is another consequence (Fermat's Little Theorem being one) of the Euclid's Proposition VII.30. As we have observed,

In algebraic terms this fact is expressed as

Indeed, n² ≡ 1 (mod p) implies that p|(n² - 1), or p|(n - 1)(n + 1). By the Proposition VII.30 either

Now, as we already remarked, multiplication by [a]_{p} permutes the set _{p}, [2]_{p}, [3]_{p}, ... [p-1]_{p}}._{p}, [3]_{p}, ..., [p-2]_{p}}

_{p}= [1]

_{p}·[2]

_{p}·[3]

_{p}· ... ·[p-1]

_{p}= [1]

_{p}·[p-1]

_{p}= [p-1]

_{p}

which is most often written as

Yet another way of writing it is *Meditationes Algebraicae* by Edward Waring. Waring ascribed the result to a student of his John Wilson. Waring gave a proof of the theorem in the third edition of *Meditationes Algebraicae* in 1782. However the first published proof was given by J. L. Lagrange yet in 1770 [Hilton, p. 41, Ore, p. 259].

For composite moduli, the Wilson's formula fails. Instead we have

*composite*integer then

Assuming n = PQ, where P ≤ Q and each greater than 1, we note that both P and Q appear as multiples in

Thus, at least in principle, Wilson's Theorem can be used to establish *primality* of a given number.

(A combinatorial proof of Wilson's theorem can be found elsewhere.)

### References

- J. H. Conway and R. K. Guy,
*The Book of Numbers*, Springer-Verlag, NY, 1996. - H. Davenport,
*The Higher Arithmetic*, Harper&Brothers, NY - U. Dudley,
*Elementary Number Theory*, Dover, 2008 - R. Graham, D. Knuth, O. Patashnik,
*Concrete Mathematics*, 2nd edition, Addison-Wesley, 1994. - P. Hilton, D. Holton, J. Pederson,
*Mathematical Reflections*, Springer Verlag, 1997 - R. Honsberger,
*Mathematical Gems II*, MAA, 1976 - O. Ore,
*Number Theory and Its History*, Dover Publications, 1976

- Modular Arithmetic
- Chinese Remainder Theorem
- Euclid's Algorithm
- Pick's Theorem
- Fermat's Little Theorem
- Wilson's Theorem
- Euler's Function
- Divisibility Criteria
- Examples
- Equivalence relations
- A real life story

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