Let dç, (n) = p„, the nth prime, for n > 1, and let dk+x(n) = \dk(n) - dk(n + 1) for k > 0, n > 1. A well-known conjecture, usually ascribed to Gilbreath but actually due to Proth in the 19th century, says that dk(\) = 1 for all k > 1. This paper reports on a computation that verified this conjecture for k < tt(1013) » 3 x 10". 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.