Iterated absolute values of differences of consecutive primes

Abstract:

Let ${d_0}(n) = {p_n}$, the nth prime, for $n \geq 1$, and let ${d_{k + 1}}(n) = |{d_k}(n) - {d_k}(n + 1)|$ for $k \geq 0,n \geq 1$. A well-known conjecture, usually ascribed to Gilbreath but actually due to Proth in the 19th century, says that ${d_k}(1) = 1$ for all $k \geq 1$. This paper reports on a computation that verified this conjecture for $k \leq \pi ({10^{13}}) \approx 3 \times {10^{11}}$. It also discusses the evidence and the heuristics about this conjecture. It is very likely that similar conjectures are also valid for many other integer sequences.