In 1770 Edward Waring announced the following theorem
by his former student John Wilson.
This beautiful result is of mostly theoretical value because it is relatively difficult to calculate
(p-1)! In contrast it is easy to calculate
ap-1, so elementary
primality tests are built using Fermat's Little
Theorem rather than Wilson's Theorem. For example,
the largest prime ever shown prime by Wilson's theorem
is most likely 1099511628401, and even with a clever approach to calculating n!, this still took
about one day on a SPARC processer. But numbers
with ten's of thousands of digits have been shown prime using a converse of Fermat's theorem in a less than an hour.
- Wilson's Theorem.
- Let p be an integer greater than one.
p is prime if and only if (p-1)! = -1
You might want to try your hand at proving the following corollary to Wilson's theorem:
n is prime if and only if
sin(((n-1)!+1)pi/n) is zero.
See Also: WilsonPrime
Related pages (outside of this work)
- R. Crandall, K. Dilcher and C. Pomerance, "A search for Wieferich and Wilson primes," Math. Comp., 66:217 (1997) 433--449. MR 97c:11004 (Abstract available)