Iterated Absolute Values of Differences of Consecutive Primes

1993/07/01 by Andrew Odlyzko · 2 citations
Mathematics · #Analytic Number Theory Research #History and Theory of Mathematics #Limits and Structures in Graph Theory #Conjecture #Mathematics #Combinatorics #Iterated function #Prime (order theory) #Integer (computer science) #Prime number #Computation #Mathematical analysis #Algorithm

paper · doi:10.2307/2152962

openalex publication_date 1993/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

Let d₀(n) = p_n, the nth 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 19th 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.

Citations

• The segmented sieve of eratosthenes and primes in arithmetic progressions to 1012
• On the distribution of primes in short intervals
• On the order of magnitude of the difference between consecutive prime numbers

Cited by

• A game with divisors and absolute differences of exponents
• Gilbreath's conjecture [wikipedia]