For a prime $p$, it is well-known that the first $p$-torsion in $\pi_i(S^3)$ appears at $i=2p$.
Naturally, one is curious about the next homotopy group:
Is there any $p$-torsion in $\pi_{2p+1}(S^3)$?
The answer for $p=2$ is trivially positive since $\pi_5(S^3)=\pi_4(S^3)=\mathbb{Z}/2$.
So let's suppose $p$ is an odd prime. By looking at the list of $\pi_i(S^3)$, I get the impression:
$\pi_{2p+1}(S^3)$ has no $p$-torsion.
Is this true? I failed to think of anything helpful.
If this is true, how can we prove it? I have no idea to start...
Generally, let the second $p$-torsion appear at $i=k$, then:
How big and how small can $k-2p$ be?
Is there any estimate of this number $k-2p$?
The $\Lambda$-algebra should be some help. (This was originally defined in the "6-author paper", with a corrected version due to Bousfield and Kan. See also Computing the Homology of the Lambda Algebra by Tangora.) It is an algebra $\Lambda$ which forms the $E_1$-term of the Adams spectral sequence converging to the stable homotopy groups of spheres, together with subalgebras $\Lambda(n)$ which, at least for $n$ odd, form the $E_1$-term of the unstable Adams spectral sequence converging to the homotopy groups of $S^n$.
$\Lambda$ is a quotient of a free noncommutative algebra on generators $\mu_i$, $i \geq 0$, and $\lambda_i$, $i > 0$, with
The degree $j$ part of $\Lambda$ or of $\Lambda(n)$ corresponds to $\pi_{n+j}(S^{n})$. The first few of the generators are $\mu_0$ in degree $0$, $\lambda_1$ in degree $2p-3$, $\mu_1$ in degree $2p-2$, $\lambda_2$ in degree $4p-5$, etc. The element $\mu_1$ in degree $2p-2$ supports a differential and hence doesn't contribute to $\pi_{n+2p-2}(S^n)$. (I haven't checked the details for this, and it should be checked.) So the first potential $p$-torsion would be in $\pi_{n+2p-3}(S^n)$. After that, $\Lambda$ has no elements in degrees between $2p-1$ and $4p-7$, so the next possible $p$-torsion would come from $\lambda_1^2$ in $\pi_{n+4p-6}$, as well terms like $\lambda_1^2 \mu_1^j$, etc., all in the same degree.