Package de.tilman_neumann.jml.primes

Class PrimeGapTest

  • All Implemented Interfaces:
    SieveCallback
    public class PrimeGapTest
extends Object
implements SieveCallback
    Find primes with relatively large prime gaps, say ratios p(i)/p(i-1) > p(k)/p(k-1) for all k > i. Bertrand’s postulate states that there is a prime in any interval [n, 2n]. Legendre's conjecture says that there is at least one prime in any interval (n^2, (n+1)^2). But for not too small i, the true maximum ratio p(i)/p(i-1) is much smaller than that. The main purpose of this class is to find the following rational sequence: Numerators are A277718 = "Bounding prime for the first k-Ramanujan prime." Denominators are A209407 = "Primes p(i) such that p(i+1)/p(i) > p(k+1)/p(k) for all k>i" Bounds are taken from Christian Axler, Thomas Leßmann: "An explicit upper bound for the first k-Ramanujan prime" 2018, https://arxiv.org/pdf/1504.05485.pdf, and the proof procedure has been derived from comments by Charles Greathouse IV. in A209407. A few results: 1. Testing all primes below 1e11 allows us to prove the first 87 sequence elements. 2. Trudgian's bound is not satisfied at a(75) ? 3. Trudgian's bound seems to be the one of most practical relevance; the transition point between Trudgian's bound and Axler/Leßmann's bound is at approximately e^131.1687 = 9.24363...*10^56.
    Author:
    Tilman Neumann

    • Constructor Detail

      • PrimeGapTest

        public PrimeGapTest​(long limit)

    • Method Detail

      • run

        public void run()

      • main

        public static void main​(String[] args)