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First, we present a new proof of Glaisher's formula dating from 1900 andconcerning Wilson's theorem modulo p^2. Our proof uses p-adic numbers andFaulhaber's formula for the sums of powers (17th century), as well as morerecent results on Faulhaber's coefficients obtained by Gessel and Viennot.Second, by using our method, we find a simpler proof than Sun's proof regardinga formula for (p-1)! modulo p^3, and one that can be generalized to higherpowers of p. Third, we can derive from our method a way to compute the Stirlingnumbers modulo p^3, thus improving Glaisher and Sun's own results from 120years ago and 20 years ago respectively. Last, our method allows to find newcongruences on convolution of divided Bernoulli numbers and convolutions ofdivided Bernoulli numbers with Bernoulli numbers.

Wilson's theorem modulo p^2 derived from Faulhaber polynomials