William wrote:OddPerfect has many tasks that we work on from time to time. The variety of easy and hard tasks keeps things interesting. Yoyo@home's BOINC ecm has already pushed several tasks beyond where I expected to get in all of 2009. In addition we've had the fun of being major contributors to Paul Zimmermann's the Top 10 Factors of 2009.

In the last couple of weeks Yoyo@home's work has made large progress on an easy task, a hard task, and a very hard task. Most work on OddPerfect numbers depends on factor chains. Factor chains use the property that if a number "x" divides an odd perfect number, then the factors of a related number y also divide the odd perfect number. These factors of y can then be used to generate still more factors - so the more factors we can find the better we can build factor chains.

The easy task is finishing partial factorization that have 120-160 digits left. The hard task is factoring numbers 135-500 digits with no known factors. The very hard task is factoring the five "most wanted" numbers.

In the easy task we extended previous work on some easy numbers. The life cycle of these numbers is several stages of searching for numbers with ecm and then turning them over to factoring by the Number Field Sieve (NFS). We pushed one set of these numbers to the next stage and found some factors in the process. In the future we will give these numbers one more stage then turn them over to NFS. There is another batch of over 100 easy numbers queued up for the first stage work, and several more batches that will be queued in the future.

In the hard task we found a lot of "first known factors." Most of our records in the top 10 factors of 2009 come from this task. We should have found about 1/3 of the 50-digit factors in these numbers. In the future we will look for 50 and 55 digit factors of these numbers.

In the very hard task yoyo@home is expected to find 1/3 of the 55 digit factors in the five "most wanted" numbers. We have finished two and are 90% done with the third - no factors found yet. We will keep returning to these numbers.

I have a couple of other tasks queued up and I have plans for still more tasks in the future. I'll tell you about these tasks as we get to them in the queue.

Thanks to all the BOINC crunchers for the great progress in these tasks.

William

yoyo