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In this paper we prove that $\beta_{p^2/p^2-1}$ survives to $E_\infty$ in theAdams-Novikov spectral sequence for $p\geqslant 5$. As an easy consequence weprove 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.

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