Some extensions in the Adams spectral sequence and the 51-stem

We show a few nontrivial extensions in the classical Adams spectral sequence. In particular, we compute that the 2-primary part of $\pi_{51}$ is $\mathbb{Z}/8\oplus\mathbb{Z}/8\oplus\mathbb{Z}/2$. This was the last unsolved 2-extension problem left by the recent works of Isaksen and the authors (\cite{Isa1}, \cite{IX}, \cite{WX1}) through the 61-stem. The proof of this result uses the $RP^\infty$ technique, which was introduced by the authors in \cite{WX1} to prove $\pi_{61}=0$. This paper advertises this method through examples that have simpler proofs than in \cite{WX1}.