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Odlyzko93

A. M. Odlyzko, "Iterated absolute values of differences of consecutive primes," Math. Comp., 61 (1993) 373-380.  MR 93k:11119

Abstract: Let d₀(n) = p_n, the n-th prime, for n ≥ 1, and let d_k+1(n) = | d_k(n) - d_k(n+1) | for k ≥ 0, n ≥ 1. A well known conjecture, usually ascribed to Gilbreath but actually due to Proth in the 19-th century, says that d_k(1) =1 for all k ≥ 1. This paper reports on a computation that verified this conjecture for k ≤ π(10¹³) ~ 3 × 10¹¹. 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.