An analogue of the Proth-Gilbreath conjecture

Let gpf (x) be the greatest prime factor of the integer x > 1. We formulate and prove an analogue of the Proth-Gilbreath's conjecture that uses the function (x, y) {mapping} gpf (x + y) instead of (x, y) {mapping} |y - x|in the definition of the recursion. Thus, the new type of recursion is induced by the mapping that associates to any infinite prime vector (q₁, q₂, q₃,...) the vector (gpf (q₁ + q₂), gpf (q₂ + q₃), gpf (q₃ + q₄),...). If we start the recursion from the initial prime vector (p₁, p₂, p₃,...), where pi represents the ith prime, then we show that the first component in any subsequent infinite vector is always an element of the special set A = {2, 3, 5, 7}. A comparative analysis of the convergence speed in the classical Proth-Gilbreath recursion versus the one in its GPF-analogue is presented. The analysis shows that the components of the iterates in the GPFanalogue of the Gilbreath recursion are rapidly decreasing and quickly become elements of A, whereas the components of the iterates in the classical Gilbreath are decreasing at a slower rate.