On the Nilpotence Order of β 1

For p > 2, β1 ∈ π 2p2−2p−2(S) is the first positive even-dimensional element in the stable homotopy groups of spheres. A classical theorem of Nishida [Nis73] states that all elements of positive dimension in the stable homotopy groups of spheres are nilpotent. In fact, Toda [Tod68] proved β 2−p+1 1 = 0. For p = 3 he showed that β 1 = 0 while β 5 1 6= 0. In [Rav86] the second author computed the first thousand stems of the stable homotopy groups of spheres at the prime 5. One of the consequences of this computation is that β 1 = 0 while β 17 1 6= 0. Our purpose here is to study the problem for larger primes. Our result is the following.