This paper proves the existence of third periodicity element in the stable homotopy of spheres ¼pn+2q+(pn-s)(p+1)q-q-3S which is of order p and is represented by rpn/s in the E32,* term of the Adams-Novikov spectral sequence. On the way of proving the main result, a new family in the stable homotopy groups of V (1) which is represented by hn in the E1;pnq 2 term of the classical Adams spectral sequence also is detected, where p≥5; q=2(p-1); n≥1 and 1≤s≤pn-1.

This paper proves the existence of an order p element in the stable homotopy group of sphere spectrum of degree p"q+p"q +q-4 and a nontrivial element in the stable homotopy group of Moore spectum of degree p"q+p"q+q-3 which are represented by h0(hmbn-1-hnbm-1) and in (h0hnhm) in the E2-terms of the Adams spectral sequence respectively., where p≥7 is a prime, n≥m+2≥4; q=2 (p-1).