TITLE: Divisibility in the homotopy groups of spheres?
QUESTION [8 upvotes]: 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 $n$?

REPLY [20 votes]: 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 $n$.