I posted this question some time ago and got a great response which I now understand. However, I am still struggling to understand the leap from (*) to (**). This is Wilsons Theorem right?
(e^x)(e^x2/2)(e^x^3/3)...=1+x+x^2+... when |x|<1.
Show that the coefficient of x^19 in the power series expansion on the LHS has the form
1/19! + 1/19 + r/s,
where 19 does not divide s. (*)
Deduce that 18!= -1(mod 19). (**)
Thanks in advance