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What is known about the existence of other pairs of spheres (such as $S^2$ and $S^3$) whose homotopy groups coincide starting from some index.

A sufficient condition for this is the existence of a fiber bundle $S^m \to S^n$ with fiber having a finite number of nonzero homotopy groups (as in the case of the Hopf fibration)

P.S. I don't know if my question has a research level. If it is not, then feel free to close it.

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It is a result from Serre's thesis that for $n\geq 3$ and a prime $p$, the first $p$-torsion in $\pi_*S^n$ occurs precisely for $* = n+2p-3$. This shows that $(m,n) = (2,3)$ is the only pair of (edit: positive integers) $m<n$ with $\pi_*S^m \cong \pi_*S^n$ for $*$ large enough.

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    $\begingroup$ There is also $S^0$ and $S^1$. $\endgroup$ Feb 20, 2022 at 18:54

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