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 $n$.
answered Nov 25, 2019 at 3:52
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6$\begingroup$ 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? $\endgroup$ Commented Nov 25, 2019 at 7:26
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$\begingroup$ I guess it is assumed that $k\ge 2$. Is there any other implicit assumption on $k$? $\endgroup$– YCorCommented Nov 25, 2019 at 10:12
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$\begingroup$ For sure $k\neq 1$, but all other $k$ are ok. $\endgroup$ Commented Nov 25, 2019 at 15:32