2020-08-09, 21:03   #775
Pascal Ochem

Apr 2006

22×23 Posts

My policy is to increase the lower bound by a factor 10^100 when the run takes less than one week.
A couple of years ago, it took 3 weeks to prove 10^2100.
The goal of the run of july 2020 was to get the timing (it took 12 days) and to update the list of roadblocks.
Some composites would have been needed in order to circumvent sigma(3^660) or sigma(732541^46) but are not needed elsewhere.
They are not needed anymore and do not appear in the updated mwrb2100.txt

Since the program was not aiming at a new lower bound, it was run with trial factoring up to 300 only,
in order to catch the tiny factors that may cause the abundancy to exceed 2 (a higher trial limit or some ECM would slow things down).
This produces a load of uninteresting factors.
Trial factoring up to 10^10 is indeed required for roadblock composites. This is checked a posteriori.

The list of useful factors has been updated:
http://www.lirmm.fr/~ochem/opn/checkfacts.txt.gz
Quote:
 Does it even matter if there are unfound small factors in non-roadblock composites apart from them causing gcd errors?
No. The program encountered over 18 million composites, attacking them all with ECM would be a big effort for low benefit.
Most probably, the cofactor will be composite.
In this case, a given factor $$P$$ with $$d$$ digits counts for $$d$$ digits if $$P$$ is in the unfactored part
and counts for $$2d$$ digits if $$P$$ is in the factored part, because we can usually assume that $$P^2$$ divides the OPN.
So finding $$P$$ contributes only $$d$$ digits to the 2100 digits, in only one or few branches of the proof tree.

The composites worth considering are in mwrb2100.txt and t2100.txt