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

“…7 for more details. Odlyzko [3] 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.…”

“…, X (1+δ)L ) such that 2 To be light on notation, we suppress ceiling and floor functions in the rest of this section. 3 Here we have abused notation, by associating the i-tuple that X j+1 , . .…”

“…Odlyzko, at the end of Section 2 of [3], speculates that such a random sequence indeed will have first term 1 from some iteration onwards. Additionally, Problem 68 of [2] asks what gap or density properties of an initial sequence suffices to ensure the conclusion of Gilbreath's conjecture.…”

Helly numbers of algebraic subsets of ℝ d and an extension of Doignon’s Theorem

“…We have been able to show that h(P 2 ) ≥ 14 and h((Z \ P) 2 ) is finite (e.g., through a simple modification of the argument used in Theorem 1.4 and Lemma 2.3). This problem appears to be related to the Gilbreath-Proth conjecture [29] on the behavior of differences of consecutive primes.…”

Gilbreath Equation, Gilbreath Polynomials, and Upper and Lower Bounds for Gilbreath Conjecture

“…, p n ); then, there are some interesting computational proofs of GC(n). R. B. Killgrove and K. E. Ralston showed GC(63419) in 1959 [2] and A. M. Odlyzko showed GC 10 13 in 1993 [3].…”