On a four-manifold, there is apparently a relation between the first Pontryagin class modulo 4 and the Pontryagin square of the second Stiefel-Whitney class:


$\mathfrak{P}(w\_2) = p\_1 \; {\rm mod} \; 4$


This fact is for instance mentioned in the comments of [this question](https://mathoverflow.net/questions/61043/the-stiefel-whitney-and-pontryagin-classes-of-so3-bundles), but I have been unable to find a proof of it. 


My question is: Is it true that on an 8-manifold, the analogous relation 


$\mathfrak{P}(w\_4) \stackrel{?}{=} p\_2 \; {\rm mod} \; 4$


holds? 



Pontryagin square of Stiefel-Whitney classes and Pontryagin classes
In your first claim, it is a bit unclear what bundle you are considering. It is false for the tangent bundle of $\mathbb{C}P^2$: $\mathfrak{P}(w\_2) = c\_1^2 = 9 \; {\rm mod} \; 4$, while $p\_1 = 3$. The context of the question you link to is arbitrary oriented rank 3 bundles over a 4-dimensional base. Is that what you have in mind?


The Pontrjagin squares in the cohomology of $BO$ were first computed by Wu in
*On Pontrjagin classes. III* ([MR0115179](http://www.ams.org/mathscinet-getitem?mr=0115179)).
It is not so easy to get hold of a digital copy of the paper or its
translation, but one of the main results is Theorem 5 of Section 4:


$$\mathfrak{P}(w\_{2i+1}) = \beta\_4 Sq^{2i}w\_{2i+1} + \theta\_2(w\_1
Sq^{2i}w\_{2i+1})$$
$$\mathfrak{P}(w\_{2i}) = \rho\_4 p\_i + \beta\_4(w\_{2i-1}w\_{2i}) + \theta\_2
\left(w\_1 Sq^{2i-1}w\_{2i} + \sum\_{j=0}^{i - 1} w\_{2j}w\_{4i-2j}\right)$$
where $\rho\_4$ is mod 4 reduction of coefficients, $\beta\_4$ is the mod
4 reduction of the Bockstein map $H^i(-;Z/2) \to H^{i+1}(-; Z)$, and
$\theta\_2 : H^i(-;Z/2) \to H^i(-;Z/4)$ is induced by the inclusion $Z/2
\to Z/4, x \mapsto 2x$.


As a special case, we find that for any orientable vector bundle (i.e. $w\_1 = 0$)
$$ \mathfrak{P}(w\_2) = \rho\_4 p\_1 + \theta\_2(w\_4) . $$


It's worth pointing out the corollary that Wu deduces from the above calculation (and that Stiefel-Whitney classes are homotopy invariants by Wu's formula): the mod 4 reductions of the Pontrjagin classes of a closed smooth manifold are homotopy invariants.


