All of the following computations were performed on a late 2015 Dell XPS 8900 personal computer with a 64-bit Intel Core I7 processor @ 4.0GHz with 16GB of DDR2 RAM.
Factorization of Six Fibonacci Sequence Numbers:
Fibonacci 500
# digits 105
5 ^ 2 p # digits 1
15 c # digits 2
101 p # digits 3
401 p # digits 3
1661 c # digits 4
3001 p # digits 4
10291 c # digits 5
570601 p # digits 6
112128001 p # digits 9
1353439001 p # digits 10
28143378001 p # digits 11
5465167948001 p # digits 13
84817574770589638001 p # digits 20
158414167964045700001 p # digits 21
Runtime (s) = 1.206000
Fibonacci 505
# digits 106
5 p # digits 1
743519377 p # digits 9
44614641121 p # digits 11
770857978613 p # digits 12
960700389041 p # digits 12
12588421794766514566269164716286291055826556238643852856601641 p # digits 62
Runtime (s) = 1.959000
Fibonacci 510
# digits 107
2 ^ 3 p # digits 1
11 p # digits 2
61 p # digits 2
1021 p # digits 4
1597 p # digits 4
3469 p # digits 4
3571 p # digits 4
9521 p # digits 4
53551 p # digits 5
95881 p # digits 5
142445 c # digits 6
1158551 p # digits 7
3415914041 p # digits 10
20778644396941 p # digits 14
20862774425341 p # digits 14
81358225616651 c # digits 14
162716451241291 p # digits 15
Runtime (s) = 2.682000
Fibonacci 515
# digits 108
5 p # digits 1
519121 p # digits 6
5644193 p # digits 7
512119709 p # digits 9
84388938382141 p # digits 14
300367026458796424297447559250634818495937628065437243817852436228914621 p # digits 72
Runtime (s) = 7.861000
Fibonacci 520
# digits 109
131 p # digits 3
451 c # digits 3
521 p # digits 3
2081 p # digits 4
2161 p # digits 4
3121 p # digits 4
24571 p # digits 5
90481 p # digits 5
2519895 c # digits 7
21183761 p # digits 8
57089761 p # digits 8
102193207 p # digits 9
1932300241 p # digits 10
14736206161 p # digits 11
5836312049326721 p # digits 16
42426476041450801 p # digits 17
Runtime (s) = 5.155000
Fibonacci 525
# digits 110
2 p # digits 1
5 p # digits 1
421 p # digits 3
701 p # digits 3
3001 p # digits 4
3965 c # digits 4
4201 p # digits 4
141961 p # digits 6
2553601 p # digits 7
230686501 p # digits 9
8288823481 p # digits 10
82061511001 p # digits 11
19072991752501 c # digits 14
8481116649425701 p # digits 16
17231203730201189308301 p # digits 23
Runtime (s) = 2.026000
Factorization of Six Lucas Sequence Numbers
Lucas 340
113709744839525149336680459091826532688903186653162057995534262332121127
# digits 72
7 p # digits 1
2161 p # digits 4
5441 p # digits 4
897601 p # digits 6
23230657239121 p # digits 14
17276792316211992881 p # digits 20
3834936832404134644974961 p # digits 25
Runtime (s) = 109.103000
Lucas 345
# digits 73
2 ^ 2 p # digits 1
31 p # digits 2
461 p # digits 3
1151 p # digits 4
1529 c # digits 4
324301 p # digits 6
686551 p # digits 6
1485571 p # digits 7
4641631 p # digits 7
19965899801 c # digits 11
117169733521 p # digits 12
3490125311294161 p # digits 16
Runtime (s) = 0.032000
Lucas 350
13985374084677485786380981408251904922622980674054858121032362563653278123
# digits 74
3 p # digits 1
401 p # digits 3
2801 p # digits 4
11521 c # digits 5
28001 p # digits 5
570601 p # digits 6
12317523121 p # digits 11
248773766357061401 p # digits 18
7358192362316341243805801 p # digits 25
Runtime (s) = 21.047000
Lucas 355
69362907070206748494476200566565775354902428015845969798000696945226974645
# digits 74
5 p # digits 1
4261 p # digits 4
6673 p # digits 4
75309701 p # digits 8
309273161 p # digits 9
46165371073 p # digits 11
9207609261398081 p # digits 16
49279722643391864192801 p # digits 23
Runtime (s) = 40.726000
Lucas 360
769246427201094785080787978422393713094534885688979999504447628313150135520
# digits 75
2 ^ 5 p # digits 1
3 ^ 2 p # digits 1
23 p # digits 2
41 p # digits 2
105 c # digits 3
107 p # digits 3
241 p # digits 3
2161 p # digits 4
2521 p # digits 4
3439 c # digits 4
8641 p # digits 4
20641 p # digits 5
103681 p # digits 6
109441 p # digits 6
191306797 c # digits 9
10783342081 p # digits 11
13373763765986881 p # digits 17
Runtime (s) = 0.032000
Lucas 365
19076060504701386559675231910437330047906343529583769121365013189782992678011
# digits 77
11 p # digits 2
151549 p # digits 6
514651 p # digits 6
7015301 p # digits 7
8942501 p # digits 7
9157663121 p # digits 10
11899937029 p # digits 11
3252336525249736694804553589211 p # digits 31
The following two numbers were first factorized by J. M. Pollard on an 8-bit Phillips P2012 personal computer with 64 KB RAM and two 640 KB disc drives. The times required by Pollard were 41 and 47 hours.
2^144-3
22300745198530623141535718272648361505980413
# digits 44
492729991333 p # digits 12
45259565260477899162010980272761 p # digits 32
Runtime (s) = 0.086000
2^153+3
11417981541647679048466287755595961091061972995
# digits 47
5 p # digits 1
11 p # digits 2
600696432006490087537 p # digits 21
345598297796034189382757 p # digits 24
Runtime (s) = 0.676000
Partial factorization of the Twelfth Fermat Number 2^4096+1
# digits 1234
114689 p # digits 6
26017793 p # digits 8
63766529 p # digits 8
190274191361 p # digits 12
Runtime (s) = 1532.878000
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 