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Fermat Quotients

Preannounced Status Continues

Welcome. Feel free to poke around. But we're not really ready to open yet. Before we open, there a couple of programs I want to have available for download. These programs will make it easy to search for odd perfect numbers in personal niches, to add factors that extend these personal searches, and to consolidate the new factors among a community. In the meantime, I've been doing a manual search for road blocks, running an ECM server at oddperfect.no-ip.com:8201, and accepting individual factorizations from the composites page. There is also an ECM server with smaller assignments at port 8202.


We are searching for odd perfect numbers. Our main project will be to increase the lower bound beyond the 10300 limit in the Brent, Cohen, & te Riele (BCR) paper. While waiting for difficult factorizations in that project, we will pursue side projects such as searching with limited components. If the limits are reachable by QS or personal SNFS, the side searches may reach thousands of digits.

Our major approach will be community ECM factorization of road block numbers. An ECM server is already available at oddperfect.no-ip.com port 8201. This is a standard ECMNET server - if you need the client program it is available from ElevenSmooth. After sufficient ECM, composites of greater interest will be factored by Number Field Sieve.

News: 10 Primes, 1500 digits and growing

Ryan Propper has been factoring many roadblocks, paving the way for raising the bounds in OPN proofs. Particularly notable is the recent factoring of 547107-1, a C291 with no previously known factors, into P60 * P232.

Pace Nielsen has a preprint #22 that shows, among other things, that OPNs have at least 10 distinct prime divisors

Pascal Ochem and Michael Rao are orchestrating an effort to raise the lower bound of an Odd Perfect Number. They have announced completion to 1500 digits and are continuing to advance. You can follow progress at Pascal's site.

There are thousands of incomplete factorizations in the factor chains. Complete factorizations are not necessary, but their absence interferes with the aesthetics and efficiency of the proofs because factor chains are grown with smaller factors. In the short run these smaller factors lead to longer chains. Even more problematic, when factorizations are later found, the subchains based on the shorter factors become unnecessary. Pascal's orchestration includes a comprehensive list of the composites that can cause such obsolescence. Each of these factorizations makes the proof shorter and improves the efficiency of future extensions. Many people have contributed to Pascal Ochem's list of first roadblocks.

Status: Smallest Known Road Blocks

These are the known road blocks to a traditional proof under 10600. Methods are known to exceed these bound without these factorizations, but the resulting higher bounds are usually limited by these same unknown factorizations.

An additional base is needed to extend the BCR proof above 10473. The selection of 61 will complete the proof and adds no known road blocks until 10602.

Blocked At Chain Comments
10597 σ(5) = 2 * 3
σ(32) = 13
σ(13268) = C299
ECM progressing at 26e7 (60 digits).

Plans for Variations

Road blocks are usually caused by the inability to factor a large number which would put most of the limiting value into a single component. Rather than a single massive component, it seems likely that an odd perfect number will have many moderate sized components. We can search to much higher bounds if we include restrictions on the size of individual components. For example, the roadblock at 10478 could be circumvented by searching for odd perfect numbers with no component exceeding 10232, or with no exponent exceeding 100. Introducing limits of this style creates a vast accessible, customizable search space. OddPerfect.org will create tools to search user selected customizations and will be a clearing house for the factoriztions found in these searches.

The Arguments Against Existence

I have put together pages considering the arguments that odd perfect number do not exist. The main page addresses Sylvester's Web of Conditions. There is a separate page devoted to the Pomerance Heuristic.

Small Composites

Composites suitable for Quadratic Sieve and/or Personal SNFS, although not strictly necessary for the Odd Perfect Number Search, are listed on the composites page.