Iterated Absolute Values of Differences of Consecutive Primes

Abstract

Abstract. 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.

A random analogue of Gilbreath's conjecture

“…Proth's proof was wrong. Odlyzko [2] verified Gilbreath's conjecture for 1 ≤ i ≤ π(10 13 ) ≈ 3.34 × 10 11 . One is led to wonder how special the primes are in Gilbreath's conjecture and whether any sequence beginning with 2 followed by an increasing sequence of odd numbers with small and "random" gaps between them will have first term 1 from some iteration onwards.…”

Increasing integer sequences and Goldbach's conjecture

“…e.g. [13,17]). This conjecture is best explained by an example: let us take a short prefix of the sequence of primes, say 2 3 5 7 11 13 17, and construct a triangle in which each element is the absolute value of the difference of the two elements above it, like in Figure 1a.…”

On the Gap sequence and Gilbreath's conjecture

“…The conjecture was studied long before Gilbreath's observation by Francois Proth who allegedly had obtained a proof which was invalidated [1]. The conjecture remains unresolved as of now but it had been verified computationally to be true by Andrew Odlyzko, that d k 1 = 1 for all k ≤ n = 3.4 × 10 11 in 1993 [3]. There has also been spates of attempts generalizing Gilbreath's conjecture by many authors to other non-prime sequences obeying similar distribution of prime numbers with certain specifications on their gaps [4] but various counter examples have now been found.…”