proof of Wilson’s theorem using the Wilson quotient
If is composite, then its greatest prime factor is at most , and as long as (and the smallest positive composite number is 4). Therefore, being the product of the numbers from 1 to includes among its divisors the greatest prime factor of , and indeed all its divisors. Since two consecutive integers are always coprime, it is the case that . Therefore will divide evenly but not . So the Wilson quotient will be a rational number but not an integer.
If we only meant to prove the converse of Wilson’s theorem we’d be done at this point. But we set out to prove that not only is the stated relation false for composite numbers, but that it is true for primes. Proving the former is quite easy. Proving the latter is harder, and in fact neither Edward Waring nor John Wilson left a proof. (Koshy, 2007)
If is prime, then obviously its greatest prime factor is itself, and will have only 1 as a divisor in common with . But how do we prove that the next number after is a multiple of without having to factorize several values of and hoping the proof makes itself apparent thus? Modular multiplication comes to the rescue. Since is prime, we can multiply any number from 2 to by another of the same range and the product will be congruent to 1 modulo .
So in general multiplying the range 2 to gives a number that is congruent to 1 modulo if is prime. (With composite, modular multiplication causes a zeroing out of the range’s overall product). is different, satisfying . And since , modular multiplication of the range 2 to gives . Thus when is prime, so . Therefore, , which is greater than , is also divisible by it, and thus dividing it by yields an integer. ∎
- 1 Thomas Kochy, Elementary Number Theory with Applications, 2nd Edition. London: Elsevier (2007): 321 - 323
|Title||proof of Wilson’s theorem using the Wilson quotient|
|Date of creation||2013-03-22 17:58:57|
|Last modified on||2013-03-22 17:58:57|
|Last modified by||PrimeFan (13766)|