I am working on a factorization of the Thirteenth Fermat number which is 2 ^ 8192 + 1 and is 2,467 decimal digits in length. I am using Pollard’s factoring with cubic integers on the number (2 ^ 2731) ^ 3 + 2. I am also utilizing a homegrown variant of the venerable Pollard and Brent rho method and Arjen K. Lenstra’s Free LIP Elliptic Curve Method. I can factor the seventh Fermat number 2 ^ 128 + 1 in five to thirty minutes using my C# code. The factoring with cubic integers code is in C and uses Free-LIP.
Fermat factoring status (prothsearch.com)
The following is a run of Lenstra’s ECM algorithm:
== Data Menu ==
1 Simple Number
2 Fibonacci Sequence Number
3 Lucas Sequence Number
4 Exit
Enter option (1 – 4): 1
Enter a number to be factored: 2^8192+1
Enter a random number generator seed: 1
== Factoring Menu ==
1 Lenstra’s ECM
2 Lenstra’s Pollard-Rho
3 Pollard’s Factoring with Cubic Integers
Option (1 – 3): 1
2710954639361 p # digits 13
3603109844542291969 p # digits 19
Runtime (s) = 17015.344000
I aborted the previous computation due to the fact I was curious about the number of prime factors that could be found on personal computer. I will try a lot more calculation time in a future run. My homegrown application is able to at least find the first factor of Fermat Number 13.
Author: jamespatewilliamsjr
My whole legal name is James Pate Williams, Jr. I was born in LaGrange, Georgia approximately 70 years ago. I barely graduated from LaGrange High School with low marks in June 1971. Later in June 1979, I graduated from LaGrange College with a Bachelor of Arts in Chemistry with a little over a 3 out 4 Grade Point Average (GPA). In the Spring Quarter of 1978, I taught myself how to program a Texas Instruments desktop programmable calculator and in the Summer Quarter of 1978 I taught myself Dayton BASIC (Beginner's All-purpose Symbolic Instruction Code) on LaGrange College's Data General Eclipse minicomputer. I took courses in BASIC in the Fall Quarter of 1978 and FORTRAN IV (Formula Translator IV) in the Winter Quarter of 1979. Professor Kenneth Cooper, a genius poly-scientist taught me a course in the Intel 8085 microprocessor architecture and assembly and machine language. We would hand assemble our programs and insert the resulting machine code into our crude wooden box computer which was designed and built by Professor Cooper. From 1990 to 1994 I earned a Bachelor of Science in Computer Science from LaGrange College. I had a 4 out of 4 GPA in the period 1990 to 1994. I took courses in C, COBOL, and Pascal during my BS work. After graduating from LaGrange College a second time in May 1994, I taught myself C++. In December 1995, I started using the Internet and taught myself client-server programming. I created a website in 1997 which had C and C# implementations of algorithms from the "Handbook of Applied Cryptography" by Alfred J. Menezes, et. al., and some other cryptography and number theory textbooks and treatises.
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Numerical Explorations 