Hensel′s Lemma and the Divisibility by Primes of Stirling-like Numbers

The 2-adic valuations of differences of Stirling numbers of the second kind

“…The values min{v p (k!S(n, k)) : m ≤ k ≤ n} are important in algebraic topology, see, for example, [2,6,8,9,10,17,18]. Some work evaluating v p (k!S(n, k)) have appeared in above papers as well as in [5,7,23]. Lengyel [15] studied the 2-adic valuations of S(n, k) and conjectured, proved by Wannemacker [21], that ν 2 (S(2 n , k)) = s 2 (k) − 1, where s 2 (k) means the base 2 digital sum of k. Lengyel [16] showed that if 1 ≤ k ≤ 2 n , then ν 2 (S(c2 n , k)) = s 2 (k) − 1 for any positive integer c. Hong et al [11] proved that ν 2 (S(2 n + 1, k + 1)) = s 2 (k) − 1, which confirmed a conjecture of Amdeberhan et al [1].…”

A Congruence for Factorials

“…Since this implies that Γ p (p r x)/Γ p (p r ) x ≡ 0 mod p 3r , we see that [1,Proposition 5] yields…”

“…If x 3 − x ∈ 5Z 5 , then we certainly have Γ 5 (5x) ≡ Γ 5 (5) x mod 5 4 , so that, by [1,Proposition 5],…”

“…It follows (see [1,Proposition 4]) that log Γ 2 (2x) = Γ 2 (2x) − 1 , if x ∈ 4Z 2 or x ∈ 1 + 4Z 2 , Γ 2 (2x) + 1 , if x ∈ 2 + 4Z 2 or x ∈ 3 + 4Z 2 . Now (3) holds for p = 2, which, setting x = 1, yields 0 = 2λ 0 − m 1 2 2m+1 λ m 2m(2m + 1) .…”

On $p$-adic valuations of Stirling numbers

“…The problem of computation of the padic valuations (with emphasis on 2-adic valuations) of Stirling numbers of the second kind and their relatives generated a lot of research, e.g. [7,10,11,16,20,21,26]. The Stirling numbers of the second kind appear in algebraic topology.…”