Factorizations of the Seventh Fermat Number
and Other Composite Integers Using the
Pollard-Shor-Williams C# App and the LIP
Lenstra Elliptic Curve Method
It took 20.1 hours for J. M. Pollard to
factor the Seventh Fermat Number in
December 1988. He was using an 8-bit
Philips P2012 personal computer with
64k RAM and two 640k floppy drives.
His seven programs were written in the
Pascal computer language. The current
author was using a 64-bit Core i5 Dell
Latitude 3410 notebook computer with
8 GB RAM and a 235 GB solid state
hard drive. The computer language was
Windows 32 vanilla C in the Release x64
Configuration. The operating system was
Windows 11 Pro with the Visual Studio
2022 Community Version Integrated
Development Environment and C#.
2^128+1
340282366920938463463374607431768211457 39
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 1
59649589127497217 p 17
5704689200685129054721 p 22
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:20:31.359
Function Evaluations = 661379586
2^128+1
340282366920938463463374607431768211457 39
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 2
59649589127497217 p 17
5704689200685129054721 p 22
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:12:31.146
Function Evaluations = 283327140
2^128+1
340282366920938463463374607431768211457 39
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 3
59649589127497217 p 17
5704689200685129054721 p 22
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:21:53.637
Function Evaluations = 371472150
2^128+1
340282366920938463463374607431768211457 39
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 4
59649589127497217 p 17
5704689200685129054721 p 22
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:15:25.712
Function Evaluations = 197866620
2^128+1
340282366920938463463374607431768211457 39
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 5
59649589127497217 p 17
5704689200685129054721 p 22
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:03:06.030
Function Evaluations = 19501012
2^128+1
340282366920938463463374607431768211457 39
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 6
59649589127497217 p 17
5704689200685129054721 p 22
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:12:25.076
Function Evaluations = 198404541
2^128+1
340282366920938463463374607431768211457 39
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 7
59649589127497217 p 17
5704689200685129054721 p 22
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:20:59.172
Function Evaluations = 168943987
2^128+1
340282366920938463463374607431768211457 39
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 8
59649589127497217 p 17
5704689200685129054721 p 22
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:04:58.588
Function Evaluations = 13754504
Using Lenstra's Elliptic Curve Method
enter the number to be factored below:
2^128+1
340282366920938463463374607431768211457
number has 39 digits
== Menu ==
1 Cohen's Brent-Pollard Method
2 Cohen's Trial Division
3 Lenstra's Elliptic Curve Method
4 Pollard p-1 Method First Stage
5 Pollard p-1 Both Stages
6 Pollard-Shor-Williams Method
7 Exit Application
Enter an option '1' to '7':
3
factorization is complete
Runtime in seconds:
1.00000 sec.
original number has 39-decimal digits
'c' means composite and 'p' means prime
59649589127497217 17-decimal digits p
5704689200685129054721 22-decimal digits p
enter the number to be factored below:
Miscellaneous numbers using the C# app:
2^32+1
4294967297 10
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 1
641 p 3
6700417 p 7
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:00:00.034
Function Evaluations = 57
2^41-1
2199023255551 13
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 1
13367 p 5
164511353 p 9
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:00:00.003
Function Evaluations = 1128
2^67-1
147573952589676412927 21
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 1
193707721 p 9
761838257287 p 12
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:00:00.065
Function Evaluations = 30519
2^144-3
22300745198530623141535718272648361505980413 44
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 1
492729991333 p 12
45259565260477899162010980272761 p 32
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:00:04.804
Function Evaluations = 2458746
2^32+1
4294967297 10
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 5
641 p 3
6700417 p 7
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:00:00.001
Function Evaluations = 81
2^41-1
2199023255551 13
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 5
13367 p 5
164511353 p 9
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:00:00.002
Function Evaluations = 174
2^67-1
147573952589676412927 21
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 5
193707721 p 9
761838257287 p 12
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:00:00.050
Function Evaluations = 35949
2^144-3
22300745198530623141535718272648361505980413 44
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 5
492729991333 p 12
45259565260477899162010980272761 p 32
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:00:01.613
Function Evaluations = 632928
Now Lenstra's ECM:
enter the number to be factored below:
2^32+1
4294967297
number has 10 digits
== Menu ==
1 Cohen's Brent-Pollard Method
2 Cohen's Trial Division
3 Lenstra's Elliptic Curve Method
4 Pollard p-1 Method First Stage
5 Pollard p-1 Both Stages
6 Pollard-Shor-Williams Method
7 Exit Application
Enter an option '1' to '7':
3
factorization is complete
Runtime in seconds:
0.00000 sec.
original number has 10-decimal digits
'c' means composite and 'p' means prime
641 3-decimal digits p
6700417 7-decimal digits p
enter the number to be factored below:
2^41-1
2199023255551
number has 13 digits
== Menu ==
1 Cohen's Brent-Pollard Method
2 Cohen's Trial Division
3 Lenstra's Elliptic Curve Method
4 Pollard p-1 Method First Stage
5 Pollard p-1 Both Stages
6 Pollard-Shor-Williams Method
7 Exit Application
Enter an option '1' to '7':
3
factorization is complete
Runtime in seconds:
0.00000 sec.
original number has 13-decimal digits
'c' means composite and 'p' means prime
13367 5-decimal digits p
164511353 9-decimal digits p
enter the number to be factored below:
2^67-1
147573952589676412927
number has 21 digits
== Menu ==
1 Cohen's Brent-Pollard Method
2 Cohen's Trial Division
3 Lenstra's Elliptic Curve Method
4 Pollard p-1 Method First Stage
5 Pollard p-1 Both Stages
6 Pollard-Shor-Williams Method
7 Exit Application
Enter an option '1' to '7':
3
factorization is complete
Runtime in seconds:
0.00000 sec.
original number has 21-decimal digits
'c' means composite and 'p' means prime
193707721 9-decimal digits p
761838257287 12-decimal digits p
enter the number to be factored below:
2^144-3
22300745198530623141535718272648361505980413
number has 44 digits
== Menu ==
1 Cohen's Brent-Pollard Method
2 Cohen's Trial Division
3 Lenstra's Elliptic Curve Method
4 Pollard p-1 Method First Stage
5 Pollard p-1 Both Stages
6 Pollard-Shor-Williams Method
7 Exit Application
Enter an option '1' to '7':
3
factorization is complete
Runtime in seconds:
0.00000 sec.
original number has 44-decimal digits
'c' means composite and 'p' means prime
492729991333 12-decimal digits p
45259565260477899162010980272761 32-decimal digits p
enter the number to be factored below:
For the last number 2^144-3 it took J. M. Pollard's
factoring with cubic integers 47 hours in 1988.
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.
View all posts by jamespatewilliamsjr
Numerical Explorations 