Largest known Carmichael with 3 factors

[Phil Carmody]

With a modicum of pleasure I can announce the following (and do I hope I've not made any typos, or I'll look foolish - this has had the indiginity of being transcribed by hand...) record sized 3-Carmichael. The factors have 7106, 7107 and 14206 digits apiece. I present them in the compactest way possible, which will hopefully indicate the technique I used to find them.

1*35094464013121611171*(16477#/100280245065)+1
2*35094464013121611171*(16477#/100280245065)+1
(2*35094464013121611171*(16477#/100280245065)+3)/3353805*35094464013121611171*(16477#/100280245065)+1

Where # is the primorial function, akin to the factorial, but only accumulating primes in the product.

It's a fairly boring 3-Carmichael, being
(m+1), (2m+1), (2m+3)/d.m+1
Where the division is exact. The only novelty is the size. It is the new record for all three of
a) largest smallest factor
b) largest largest factor
c) largest product (for a _3-Carmichael_, there are hugely larger Carmichaels with more factors known).

The previous record was set (almost against his will!) only a few weeks ago by Dr. David Broadhurst (4847+4847+9688 digits).

The computation had the following phases:
- half a day of pre-sieving using a handwritten sieve (using the GMP library) on my Alpha 21164/533;
- about 1 week's probable prime testing using OpenPFGW ( www.primeform.net/openpfgw/ ) on about 12 machines from PPro200s upwards (requiring favours from several friends - Andreas, Sankalp, Ravi, Joseph, thank you);
- about a day finding the divisor 'd' using OpenPFGW on a single Duron, when the first two terms had been found;
- a few seconds to prove primality of all three terms using OpenPFGW.
All in all about 28 GHz-days, and thus /very/ easy for someone else with more resources to beat!

Phil