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Old 2021-01-22, 18:11   #13
VBCurtis's Avatar
Feb 2005
Riverside, CA

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Originally Posted by mersenne1588 View Post
@VBCurtis: Hi, I hope you're doing well. Can you describe the meaning of it maybe in other words? At the moment, I have no advanced computer skills, so I don't fully understand the statement.Thank you.
Factoring numbers on Pascal Ochem's lists helps to advance the lower bound of a possible odd perfect number. There are two ways to help with this factoring list: running ECM curves to try to find small factors (say, 55 digits or smaller), or running the Number Field Sieve (NFS) on numbers that ECM fails to find a small factor for.

NFS has 3 major (and some minor, less than 5% of the job's time) phases:
1. Polynomial selection, ~5% of job time. Only needed for general NFS jobs, called GNFS; special jobs (SNFS) have polynomials as a feature of the number's structure.
2. Sieving, ~80% of job time. This step can be run on as many computers as you have available, and the data can then all be collected onto one machine before the last phase.
3. Linear Algebra, ~15% of job time. This step is best run on a single machine, so those of us running big factorization jobs like to have one fancy computer for this phase and then an army of desktops to run the longer phase 2.

There is much detail in the NFS algorithm- I am not qualified to get highly technical, but many pages have been written around here. The "factoring" subforum and the "CADO-NFS" subforum are places to browse if you get serious about these tasks. Note that you can run the software without being an expert, but it's in your best interest to start small, much smaller than the numbers in the Odd Perfect project, and to learn enough to be able to recognise when something has gone wrong or is taking far too long.

A general idea of how long these jobs take, estimated on a 6-core Haswell-era (2014 Intel) desktop:
For general numbers, GNFS: 100 digits is a few minutes, 120 digits is an hour, 140 digits is half a day, 160 digits is a week.
For special numbers, SNFS: times vary more, but 200 digits is a day or so, 260 is a week or ten days.

Once you're doing jobs that take a week, it's usually helpful to test some possible settings to see which are faster. The software packages available have good setttings for GNFS-140 digits and lower, but often 150+ digits has a lot of guesswork "from the factory". That's why it is quite helpful to practice on small jobs, to get the hang of what each phase does on screen and later to learn how to change settings and what each setting does to the data. Learning how to change settings can take months- if you run linux, CADO-NFS and its forum will be where you go to ask questions and get a guide that's much much longer than I write here.

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