Consider $SO(n)$ bundles over smooth manifolds. Then using the fact that the Stiefel-Whitney classes are the modulo 2 reductions of the Chern classes, one can prove $w_{2i}^2(E) = p_i(E) \bmod 2$. Now consider an $SO(3)$ bundle over a 4-manifold. Since the particular case I am studying concerns K3 surfaces, let us assume that $H^2(M,\mathbb{Z})$ contains no torsion. Then it is stated in various places that more is true:

$w_2^2(E) = p_1(E) \bmod 4$.

I may be missing something elementary here, but where does the mod 4 comes from?

Pontryagin square. See eom.springer.de/p/p073810.htm – BS. Apr 8 '11 at 10:40