2019/12/13 by Claire Levaillant, Levaillant, Claire
Mathematics · #05A10 #11B75 #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #G.2 #History and Theory of Mathematics #Number Theory (math.NT) #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.1912.06652
openalex publication_date 2019/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
First, we present a new proof of Glaisher's formula dating from 1900 and concerning Wilson's theorem modulo p2. Our proof uses p-adic numbers and Faulhaber's formula for the sums of powers (17th century), as well as more recent results on Faulhaber's coefficients obtained by Gessel and Viennot. Second, by using our method, we find a simpler proof than Sun's proof regarding a formula for (p-1)! modulo p3, and one that can be generalized to higher powers of p. Third, we can derive from our method a way to compute the Stirling numbers modulo p3, thus improving Glaisher and Sun's own results from 120 years ago and 20 years ago respectively. Last, our method allows to find new congruences on convolution of divided Bernoulli numbers and convolutions of divided Bernoulli numbers with Bernoulli numbers.