This section contains material relating to the discovery
and properties of the 275,977 digit prime 3*2916773 + 1
 |
|
| A linked Maple worksheet
treating the size of |
| The record for the largest known composite Fermat number - and the one which gives me the greatest personal pleasure - is still held by myself and Yves Gallot, and you may read about it in the Fermat Record Number section of my site (dating from 1999) |
The text of an email (plus, see below, two photographs of the computer screen on which the prime was found) that I sent to the Number Theory Mailing List on Wednesday, 30th May, 2001 (the official listing of this message is accessible here):
Dear colleagues,
Briefly: (3*2^916773 + 1), having 275,977 digits, has just become the largest known Proth prime, the 7th largest known prime, and a divisor of the two generalised Fermat numbers GF(916771, 3) = 3^(2^916771) + 1 and GF(916772, 10) = 10^(2^916772) + 1. The main mathematical credit goes to Yves Gallot, Paul Jobling, and George Woltman, without whom little would be possible.
Details for the interested: In July 1999 I notified you that using Yves Gallot’s (remarkable) Proth program I had found the 115,130 digit prime (3*2^382449 + 1), a factor of the Fermat number F[382447] = 2^(2^382447) + 1 (making it the largest known composite Fermat number).
I begun my search in 1999 for Proth primes of the form (3*2^n + 1), with ‘n’ in the mid 300,000’s, and have continued doing so, originally on my own, and since last October helped by a large number of non-mathematical colleagues who generously allow me idle time on their office computers. I formed the St Patrick’s College Proth-Gallot Group, and I invite you to read about it - and see our specially created logo - in the following corner of my web site http://www.spd.dcu.ie/johnbcos/proth-gallot_group_(spd).htm
Recently I began to use other remarkable and extraordinarily useful programs: Paul Jobling’s NewPGen sieve (in which he pays this tribute:
"assembler magic by Yves Gallot"), JBC (which PJ generously created, specially for me, to save my eyesight), and George Woltman’s probabilistic primality testing PRP (in which he refers to "Yves’ landmark proth.exe"). A combination of these and other factors, including:
1. a painstaking, but beneficial analysis of the Proth log files of scores of ranges of 2000 n’s has enabled me to develop a time saving strategy by hugely circumventing Yves’ cut-off point for small prime factor testing (Yves’ program sieves for small prime factors up to n^2/8)
2. generous use of extra computing facilities in my college’s small labs since the end of our teaching term has enabled me to recently take on all n’ between 702,000 and 1,050,000, all of which I expect to have completed by mid/late July.
Two days ago one of the lab computers - a 700 MHz Pentium III - running George Woltman’s PRP came up with the heart-stopping "3*2^916773 + 1 is a probable prime," and my home 1700 MHz Pentium IV - running Yves’ Proth - has just come up with:
3*2^916773 + 1 is prime ! (a = 5) [275977 digits]
3*2^916773 + 1 is prime ! (verification : a = 17) [275977 digits]
3*2^916773 + 1 divides GF(916771, 3) !
3*2^916773 + 1 divides GF(916772, 10) !
3*2^916773 - 1 ? Computing power [0.00%]
[saved at 7.41 AM, Wed 30th May 2001]
The first two of those took about 7.5 hours each, while the Fermat division test took about 22 hours. (I sat and watched the final part of the calculation, with Bach cello suites and Chinese poetry to keep me calm).
Tomorrow I will put up a digital scan photo of the n=916773 screen at my college web
site for those of you who might be interested.
Best wishes, John Cosgrave
These digital photographs were taken by my colleague Paul Murphy, a Proth-Gallot Group member:
A Maple worksheet treating the number of digits that
GF[916772, 10] has.
Click here to download the active 916773 .mws
version.
Click here to download html version 916773.html
Click here to download html version
9167731.html
Click here to download html version
916773TOC.html