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 $mathbb{Z}/8oplusmathbb{Z}/8oplusmathbb{Z}/2$. 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}.