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.

Introduction

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.

News: σ(3221⁷²) = P70 * P88 * P96

FiveMack's NFS project at the Mersenne Forum has completed the SNFS factorization σ(3221⁷²) in a project coordinated through the Mersenne Forum. The factors are

P70: 4268789244276949228762861690430742945199326935775238201407340720225441
P88: 7393720864497209547797418741987855153929288725078597776812623352014398844851509314050533
P96: 119206318557610125536465637952294120218272312333630042840751434567237099832651324905880534881861

The factorization required about 28,000 CPU-hours to gather relations and nineteen days on a quad-core AMD Phenom to combine them into a factorization. This was the smallest known roadblock for a traditional lower bound on an odd perfect number. Traditional proofs are built around a series of lemmas that there is no odd perfect number less than 10ⁿ divisible by p; this increased the value of "n" for p = 3, 5, and 11 and raised the reachable bound from 10⁵¹¹ to 10⁵⁴¹.

Status: Smallest Known Road Blocks

These are the known road blocks under 10⁶⁰⁰.

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.

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.