Consider the $k$-sphere (I'm particularly interested in $k=3$). For each positive integer $n$ let $P\_{n}$ be the set of primes that divide the order of $\pi\_{i}S^{k}$ for some $i\leq n$. 


Does the cardinality of $P\_{n}$ tend to infinity with $n$?



Divisibility in the homotopy groups of spheres?
Yes. In fact $\bigcup\_{n\geq 1} P\_n $ is the set of all primes. Serre proved that, for each odd prime $p$, there is some very predictable $p$-torsion in $\pi\_{k+(p-1)}(S^k)$, for example. 


There is a very nice theorem of McGibbon and Neisendorfer (proving, I think, a conjecture of Serre) that implies and vastly generalizes this.



> 
> **Theorem (McGibbon & Neisendorfer).** Let $X$ be a simply-connected finite complex such that $\widetilde{H}\_\*(X; \mathbb{Z}/p) \neq 0$.
>  Then $\pi\_n X$ contains $p$-torsion for infinitely many values of $n$.
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