ITERATED ABSOLUTE VALUES OF DIFFERENCES OF CONSECUTIVE PRIMES

Let d0(n) = p(n), the nth prime, for n greater-than-or-equal-to 1, and let d(k+1)(n) = \d(k)(n) - d(k)(n + 1)\ for k greater-than-or-equal-to 0, n greater-than-or-equal-to 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 greater-than-or-equal-to 1. This paper reports on a computation that verified this conjecture for k less-than-or-equal-to pi(10(13)) almost-equal-to 3 x 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.