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# Wilson's theorem

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**Wilson's theorem** concerns a relationship between factorials and prime numbers. Gottfried W. Leibniz knew of it before John Wilson and Edward Waring came along.

**Theorem W1.** (Wilson) The congruence holds true if and only if is prime. Put another way, the number is a multiple of only when is prime.

*Proof.* If is composite, then it has at least one prime factor . For , we see that . For all higher composite , we see that the number is a product that runs through every prime factor of as well as other composite numbers repeating a lot of those same prime factors, and therefore not –1.^{[1]}

The case of the prime number 2 is disposed of easily enough by examination, leaving us only the odd primes to concern ourselves with. We can also dispose of 3 by examination. The only divisors of an odd prime are 1 and itself, hence for all . For each , there is precisely one integer such that . This means that each integer from 2 to can be paired off such that their product is 1 more than a multiple of , and therefore . This falls short of the factorial, as it leaves out the pair consisting of 1 and , and we see that , and therefore satisfies the congruence specified by the theorem. □ ^{[2]}

So if we take a prime number, such as 7, we will find that the factorial of the number one less than that prime is one less than a multiple of that prime, such as is the case with 6! + 1 = 721, which is 103 times 7 (see A060371 and A007619).

There are of course many ways to prove the theorem. Decades ago, Bolker gave a proof using the algebra of polynomials and invoking Fermat's little theorem.^{[3]}

## See also

## References

- ↑ Some textbooks state the theorem in reference to prime numbers only, and thus don't bother to prove the relationship doesn't hold for composite .
- ↑ Ivan Niven & Herbert S. Zuckerman,
*An Introduction to the Theory of Numbers*. New York: John Wiley & Sons (1980): p. 34, Theorem 2.10. The proof given concerns itself only with prime. - ↑ Ethan D. Bolker,
*Elementary Number Theory: An Algebraic Approach*. Mineola, New York: Dover Publications (1969, reprinted 2007): p. 28, Theorem 13.1.