What happened to the M47?

[Batalov]

WOT rules!

[ATH]

Yes, WoT is a really great series.

[Xyzzy]

[GP2]

Originally Posted by NBtarheel_33

In the M47 announcement, note that it is "almost 10 years" between the discovery of M47 and its proof of place

There has probably never been a time when the known Mersenne primes were all known to be in the right order from 1 to N, with no undiscovered primes smaller than the Nth. At least not since Euler's time, if ever.

Earlier discoverers left gaps, didn't do double-checking 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.

[gd_barnes]

Originally Posted by GP2

In 1963, Gillies discovered M9689, M9941, M11213, however his residue for
M12143 was later found to be incorrect (by Tuckerman).

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?

[GP2]

Originally Posted by gd_barnes

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?

Back then, there were often gaps of several years between investigators trying to find Mersenne primes. M11213 was discovered by Gillies in 1963, and M19937 was discovered by Tuckerman in 1971, and there doesn't seem to be any record of anyone doing Mersenne prime searches between those two years.

So Tuckerman double-checked 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 double-checked up to 21,000 in 1979 and independently rediscovered M23,209 a couple of weeks after Noll. So they double-checked 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 double-checked the previous researcher's range without discovering a new prime themselves and thereby retriggering the need for more double-checking.

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 double-checked 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.