2021/04/25 by Levaillant, Claire I.
#05A05 #05A10 #11B75 #11Y99 #Combinatorics (math.CO) #FOS: Mathematics #G.2.1 #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2104.12264
Wilson's theorem for the factorial got generalized to the moduli p2 in 1900 and p3 in 2000 by J.W.L. Glaisher and Z-H. Sun respectively. This paper which studies more generally the multiple harmonic sums H\lbrace s\rbrace2l=1;p-1,2≤ 2l≤ p-1 modulo p4 in association with the Stirling numbers [ p
2s-1], 2≤ 2s≤ p-1 modulo p4 is concerned with establishing a generalization of Wilson, Glaisher and Sun's results to the modulus p4. We also break p-residues of convolutions of three divided Bernoulli numbers of respective orders p-1, p-3 and p-5 into smaller pieces and generalize some results of Sun for some of the generalized harmonic numbers of order p-1 modulo p4.