On the products \(\beta_ s\beta_ t\) in the stable homotopy groups of spheres (Q749919)

Let p (\(\geq 5)\) be a prime. In the p-component of the stable homotopy groups \(\pi_*(S)\) of spheres, the \(\beta\)-family \(\{\beta_ s\}_{s\geq 1}\) satisfies the relation \(uv\beta_ s\beta_ t=st\beta_ u\beta_ v\) for \(s+t=u+v\). This implies that \(\beta_ s\beta_ t=0\) if \(p| st\). The author shows that: Let s and t be positive integers with \(p\nmid st\). Then \(\beta_ s\beta_ t\neq 0\) holds in \(\pi_*(S)\) for \(s+t\in \{kp^ i-(p^{i-1}-1)/(p-1)|\) \(i\geq 1\), \(p\nmid k+1\}\).