  Let $m, n, k$ and $c$ be positive integers. Let $\nu_2(k)$ be the 2-adic
valuation of $k$. By $S(n,k)$ we denote the Stirling numbers of the second
kind. In this paper, we first establish a convolution identity of the Stirling
numbers of the second kind and provide a detailed 2-adic analysis to the
Stirling numbers of the second kind. Consequently, we show that if $2\le m\le
n$ and $c$ is odd, then $\nu_2(S(c2^{n+1},2^m-1)-S(c2^n, 2^m-1))=n+1$ except
when $n=m=2$ and $c=1$, in which case $\nu_2(S(8,3)-S(4,3))=6$. This solves a
conjecture of Lengyel proposed in 2009.
