On the P-Adic Valuations of Stirling Numbers of the Second Kind

Abstract

In this paper, we introduced certain formulas for p-adic valuations of Stirling numbers of the second kind S(n, k) denoted by vp(S(n, k)) for an odd prime p and positive integers k such that n ≥ k. We have obtained the formulas, vp(S(n, n − a)) for a = 1, 2, 3 and vp(S(cpn, cpk )) for 1 ≤ c ≤ p − 1 and primality test of positive integer n. We have presented the results of vp(S(p2, kp)) for 2 ≤ k ≤ p − 1, 2 < p < 100 and a table of vp(S(p, k)). We have posed the following conjectures from our analysis:
 
 
 1. Let p ≠ 7 be an odd prime and k be an even integer such that 0 < k < p − 1. Then vp(S(p2, kp))-vp(S(p2, p(k+1)) = 3
 
 2. If k be an integer such that 1 < k < p − 1, then the p-adic valuations satisfy vp(S(p2, kp)) = 5 or 6, if k is even; 2 or 3, if k is odd
 for any prime p > 7.
 
 
 3. For any primes p and positive integer k such that 2 ≤ k ≤ p − 1, then vp(S( p, k )) ≤ 2.