April 2013 standalone factorization of Fermat(9)

In Spring 1991, I was able to use a home-made [kludge] Arbitrary Integer Arithmetic called “Cruncher” to expand F(9) in decimal, and check Lenstra’s 1989 factorization, which I had seen published in Discovery magazine.

I wrote it in GFA Basic, on an Atari 520ST, with (after borrowing SIMM’s from friends) 4MB of RAM. It had a Motorola 68030, screaming along at 20 M_Hz. Even so, it took about 5 mins, just to expand the product. At the time, I was proud to better an IBM clone.

I used the Trachtenburg method, in decimal, to multiply, and long division (again in decimal,) for modular division, as a check. Addition and subtraction were intermediate expedients.

Upon consideration, I concluded much later that the reason Lestra didn’t put the full decimal expansion on the blackboard, might have been that the reigning Arbitrary Integer Arithmetic at the time, Mathematica, stopped at a 100 digit decimal value. Thus the factors, 7, 49 and 99 digits respectively, were available from binary, but the 155 digit product might not have been. (Fermat numbers are excessively simple in binary.)

More recently, starting April 12th, 2013, I was able to use Prime95, (ECM2 feature) to factor F(9) in just under seven (7) days, on a standalone Windows 7/Ivy Bridge PC. I used an overclocked Core i5 3470k. I stopped when the 49 digit factor broke, since I was already certain the (99 digit) co-factor was prime.

I thought I was the first to do so on a standalone machine, and I published screen shots on my now defunct web site, http://www.indenturedgeek.com and my Social Networking g+ feed, under the handle r159753j. I no longer control the password to that account, but an interested party could scrape the photos, to verify.


About James Johnson

I am an amateur mathematician & political theorist who enjoys (occasionally cerebral) humor.
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