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(cp , cp)) for 1 ≤ c ≤ p − 1 and primality test of positive integer n. We have presented the results of vp(S( p , 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 2 2 ( ( , )) ( ( , ( 1)) 3. p p v S p kp v S p p k − + = 2. If k be an integer such that 1 < k < p − 1, then the p-adic valuations satisfy 2 5 or 6, is even ( ( , )) 2 or 3, is odd p if k v S p kp if k \uf8f1 = \uf8f2 \uf8f3 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.