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

## TL;DR

This paper studies the 2-adic valuations of Stirling numbers of the second kind, confirming two conjectures related to their divisibility properties and binary digit sums.

## Contribution

It proves two conjectures by Amdeberhan, Manna, and Moll regarding 2-adic valuations of Stirling numbers of the second kind.

## Key findings

- Established the condition for equality of valuations $v_2(S(4i, 5))$ and $v_2(S(4i+3, 5))$
- Proved that $v_2(S(2^n+1, k+1))= s_2(n)-1$ for all positive integers $n$
- Confirmed conjectures on divisibility and binary digit sum relations of Stirling numbers

## Abstract

In this paper, we investigate the 2-adic valuations of the Stirling numbers $S(n, k)$ of the second kind. We show that $v_2(S(4i, 5))=v_2(S(4i+3, 5))$ if and only if $i\not\equiv 7\pmod {32}$. This confirms a conjecture of Amdeberhan, Manna and Moll raised in 2008. We show also that $v_2(S(2^n+1, k+1))= s_2(n)-1$ for any positive integer $n$, where $s_2(n)$ is the sum of binary digits of $n$. It proves another conjecture of Amdeberhan, Manna and Moll.