20180512, 23:52  #12 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
3^{2}·1,117 Posts 
WOT rules!

20180513, 00:07  #13 
Einyen
Dec 2003
Denmark
2×17×101 Posts 
Yes, WoT is a really great series.

20180514, 22:51  #14 
Aug 2002
2·7·13·47 Posts 

20180517, 21:11  #15  
Sep 2003
3·863 Posts 
Quote:
Earlier discoverers left gaps, didn't do doublechecking systematically or at all, and had some erroneous results. Guy Haworth's article from 1987 (looong PDF file) is a good overview. In 1876 Lucas calculated that M127 was prime, but several smaller exponents remained unknown, including M101, M107, M109. In 1913, Fauquembergue claimed to have proven that M101 and M137 and some others were composite, but his residues were later found to be incorrect (by Robinson). In 1952, Robinson discovered six Mersenne primes from M521 to M2281, but his residue for M1889 was later found to be incorrect (in 1961 by Hurwitz). In 1957, Riesel discovered M3217, however his residues for M2957, M2969, M3049, M3109 were later found to be incorrect. In 1961, Hurwitz discovered M4253 and M4423, however his residues for M3637, M3847, M4397, M4421 were later found to be incorrect. In 1963, Gillies discovered M9689, M9941, M11213, however his residue for M12143 was later found to be incorrect (by Tuckerman). In 1979, Nelson & Slowinski found M44497, however their residues were incorrect for about a dozen or more exponents. Slowinski notoriously did not search systematically, and missed M110513. And obviously Primenet by its design does not test exponents in monotonic rising order. Sooner or later, we will hit a large gap between exponents, and in that case, we might actually double check all exponents below M50 before we find M51. For example, there were six cases where the next exponent was more than double the previous one (127–521, 607–1279, 4423–9689, 216091–756839, 1398269–2976221, 3021377–6972593). If that turns out to be the case, then for the first time in centuries or possibly ever, the known Mersenne primes will be in known sequential order. 

20180612, 20:12  #16 
"Gary"
May 2007
Overland Park, KS
3·31·127 Posts 
Interesting analysis. But this one looks like a possible exception. The incorrect residue for M12143 is larger than the primes that Gillies discovered. At that point in time could all Mersenne's have been proven to be in the correct order? The next prime was M19937. Could there have been enough time to doublecheck everything below M11213 before M19937 was discovered?

20180612, 21:45  #17  
Sep 2003
101000011101_{2} Posts 
Quote:
So Tuckerman doublechecked Gillies' ranges, but found a new prime himself. He searched up to 21,000. Then in 1978 Nickel & Noll started searching from 21,000 to 24,500 and eventually found M21,701 and M23,209. Nelson & Slowinski doublechecked up to 21,000 in 1979 and independently rediscovered M23,209 a couple of weeks after Noll. So they doublechecked Tuckerman's work, but in the meantime Noll had found a new prime. So I don't know if there was ever a time when someone completely doublechecked the previous researcher's range without discovering a new prime themselves and thereby retriggering the need for more doublechecking. Maybe when Noll and Nelson & Slowinski independently discovered M23,209 within a couple of weeks of each other, there might have been a point when things might have been doublechecked up to that point. But then again Noll inadvertently forgot to publish a residue for M22501... Anyways, you can read Haworth's article until your eyes glaze over, but in the end it's hard to pin things down exactly. 

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