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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 $n$?

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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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    $\begingroup$ Taking $Q_n$ be the set of primes that divide the order of $\pi_nS^3$, do we also know whether the cardinality of $Q_n$ goes to infinity? $\endgroup$ – Lennart Meier Nov 25 '19 at 7:26
  • $\begingroup$ I guess it is assumed that $k\ge 2$. Is there any other implicit assumption on $k$? $\endgroup$ – YCor Nov 25 '19 at 10:12
  • $\begingroup$ For sure $k\neq 1$, but all other $k$ are ok. $\endgroup$ – Jeff Strom Nov 25 '19 at 15:32

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