A random analogue of Gilbreath’s conjecture

A well-known conjecture of Gilbreath, and independently Proth from the 1800s, states that if a0, n=pn denotes the nth prime number and ai, n=|ai-1, n-ai-1, n+1| for i, n≥1, then ai, 1=1 for all i≥1. It has been postulated repeatedly that the property of having ai, 1=1 for i large enough should hold...