Dec 31, 2017 • Dr. Tom Nicely • post

The Gapcoin network (Jonnie Frey, developer), a Bitcoin derivative which employs a hashing algorithm to search for prime gaps of high merit, has discovered a new prime gap of maximum known merit, a gap of G=8350 following the 87-digit prime:

P1=293703234068022590158723766104419463425709075574811762098588798217895728858676728143227

The merit *M=G/ln(P1)* of this gap is M=41.93878373153988, the largest merit of any known prime gap, and the first prime gap to be discovered with a merit exceeding 40. The endpoints of the gap have been certified as primes deterministically, using the Akiyama-Kida-O’Hara UBASIC implementation (1988-1992) of the APRCL2 test, due to Adleman, Pomerance, Rumely, Cohen, H. W. Lenstra, and A. K. Lenstra (1984-1987).

However, Bertil Nyman’s maximal gap of 1132, following the prime 1693182318746371 (discovered 24 January 1999), continues to exhibit the greatest known value (0.92063858855742) of the Cramér-Shanks-Granville ratio *G/ln²(p_1)*; this ratio is 0.210642105494715467 for the new Gapcoin gap. The limit superior of this ratio has been conjectured to be unity (or some even larger value); see the discussion in “New prime gaps between 1e15 and 5e16”.

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