The secondary periodic element $\beta_{p^2/p^2-1}$ and its applications

In this paper we prove that $\beta_{p^2/p^2-1}$ survives to $E_\infty$ in the Adams-Novikov spectral sequence for $p\geqslant 5$. As an easy consequence we prove that $\beta_{sp^2/j}$ are perminent cycles for all $s\geqslant 1$, $j\leqslant p^2-1$. From the Thom map $\Phi: Ext^{s,t}_{BP_*BP}(BP_*, BP_*)\longrightarrow Ext^{s,t}_A(\mathbb{Z}/p, \mathbb{Z}/p)$, we also see that $h_0h_3$ survives to $E_\infty$ in the classical Adams spectral sequence.