A megagap with merit 25.9

Document created 8 March 2013

After a suggestion in http://tech.groups.yahoo.com/group/primeform/message/11340 Pierre Cami, Michiel Jansen and Jens Kruse Andersen have found a prime gap of 1113106 from 587*43103#/2310-455704 to 587*43103#/2310+657402.

The gap ends are proven primes with 18662 digits. n# (called n primorial) is the product of all primes ≤ n. The form increased the chance of a large gap compared to random numbers.
Primality proofs were made with Primo. The gap has merit 25.90 and is the first known "megagap" (above 1 million) satisfying the requirement at Top-20 gaps with merit above 20. It is also the first megagap with proven end points.

The merit of a prime gap is defined as (gap size)/log(gap start). By the prime number theorem, this gap is around 25.9 times larger than the average gap for primes of that size. Sieving was done with APTreeSieve, a C program using the GMP library. PrimeForm/GW made 3-prp tests of the unfactored numbers.

The prime factors up to 43103 are easy to reproduce. The factors from 43117 to 10^11 are in gap1113106factors.zip. They have been verified by an independent program. PrimeForm/GW has made multiple Fermat 3-prp tests on each number with no factor below 10^11. The 64-bit residues are in gap1113106residues.zip. All residues matched in pfgw32 and pfgw64 runs on 3 different computers varying between no -a parameter and -a1, -a2, -a3. Different -a parameters choose a different sized FFT and therefore make different computations of the same residue. Dana Jacobsen has also made an independent verification of the composites with a C program using the GMP library to make BPSW primality tests.

The gap was originally announced 8 March 2013 with prp's as gap ends. The Primo certifications took 94 days and 86 days on an Intel i7-3960X on an ASUS Sabertooth X79 motherboard boosted at 4.50 GHz. The certificates are linked at the Primo Top-20 where they are the 2nd and 3rd largest as of October 2013, only 4 digits below the largest.

Links about prime gaps:
Thomas R. Nicely's First occurrence prime gaps: http://www.trnicely.net/gaps/gaplist.html
    and http://www.trnicely.net/gaps/gaps2.html#HugeGaps
Jens Kruse Andersen's
    The Top-20 Prime Gaps: http://primerecords.dk/primegaps/gaps20.htm
    First known prime megagap: http://primerecords.dk/primegaps/megagap.htm
    Largest known prime gap: http://primerecords.dk/primegaps/megagap2.htm
    New largest known prime gap: http://primerecords.dk/primegaps/megagap3.htm
Chris Caldwell's Prime Pages, The Gaps Between Primes: http://primes.utm.edu/notes/gaps.html
Eric Weisstein's Mathworld, Prime Gaps: http://mathworld.wolfram.com/PrimeGaps.html

This page was last updated 29 October 2013.
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