submitted to the Journal Nature

An Alternative to Mersenne Primes:

Finding the Largest Prime Using Convergent Sequences of Primitive Pythagorean Triples

2--(7 million zeroes)--2--(7 million zeros)--1

by Richard Allen Brown and Dr. John McKee

1248 Insititute, Charleston and Santa Barbara

Powered by Mathematica

Summary

Table 1. The first 10,000 Pythagorean Triples Coverging to an angle of ZERO
Sequence	Small Side	Diagonal	Prime Factors
n	k(n)=2*n+1	m(n)=2n2 + 2n + 1
1	3	5	prime 1/1
2	5	13	prime 2/2
3	7	25	5*5
4	9	41	prime 3/4
5	11	61	prime 4/5
6	13	85	5*17
7	15	113	prime 5/7
8	17	145	5*29
9	19	181	prime 6/9
10	21	221	13*17
11	23	265	5*53
12	25	313	prime 7/12
13	27	365	5*73
14	29	421	prime 8/14
15	31	481	13*37
16	33	545	5*109
17	35	613	prime 9/17
18	37	685	5*137
19	39	761	prime 10/19
20	41	841	29*29
21	43	925	5537
22	45	1013	prime 11/22
23	47	1105	5*221
24	49	1201	prime 12/24
25	51	1301	prime 13/25
26	53	1405	5*281
27	55	1513	17*89
28	57	1625	555*13
29	59	1741	prime 14/29
30	61	1861	prime 15/30
31	63	1985	5*397
32	65	2113	prime 16/32
33	67	2245	5*449
34	69	2381	prime 17/34
35	71	2521	prime 18/35
36	73	2665	51341
37	75	2813	29*97
38	77	2965	5*593
39	79	3121	prime 19/39
40	81	3281	17*193
41	83	3365	5*673
42	85	3613	prime 20/42
43	87	3785	5*757
44	89	3961	13*233
45	91	4141	41*101
46	93	4325	55173
47	95	4513	prime 21/47
48	97	4705	5*941
49	99	4901	131329
50	101	5101	prime 22/50
51	103	5305	5*1061
52	105	5513	37*149
53	107	5725	55229
54	109	5941	13*457
55	111	6161	61*101
56	113	6385	5*1277
57	115	6613	17*389
58	117	6845	53737
59	119	7081	73*97
60	121	7321	prime 23/60
61	123	7565	51789
62	125	7813	13*601
63	127	8065	5*1613
64	129	8321	53*157
65	131	8581	prime 24/65
66	133	8845	52961
67	135	9113	13*701
68	137	9385	5*1877
69	139	9661	prime 25/69
70	141	9941	prime 26/70
71	143	10225	55409
72	145	10513	prime 27/72
73	147	10805	5*2161
74	149	11101	17*653
75	151	11401	13*877
76	153	11705	5*2341
77	155	12013	41*293
78	157	12325	5517*29
79	159	12641	prime 28/79
80	161	12961	13*997
81	163	13285	5*2657
82	165	13613	prime 29/82
83	167	13945	5*2789
84	169	14281	prime 30/84
85	171	14621	prime 31/85
86	173	14965	54173
87	175	15313	prime 32/87
88	177	15665	513241
89	179	16021	37*433
90	181	16381	prime 33/90
91	183	16745	517197
92	185	17113	109*157
93	187	17485	513269
94	189	17861	53*337
95	191	18241	172937
96	193	18625	555*149
97	195	19013	prime 34/97
98	197	19405	5*3881
99	199	19801	prime 35/99
100	201	20201	prime 36/100

Table 2. The First 31 Primitive Pythagorean Triples of the Form n= 10**k, for all k = 1 through 31
Definition: 20000200001 = 2<4>2<4>1
k	Sequence	small side	large side	diagonal
k	n=10**k	p(n)=2*n+1	q(n)=2n2+2n	r(n)=2n2+2n+1
0	1	3	4	5 Prime
1	10	21	220	221=13*17
2	100	201	20200	20201 Prime
3	1000	2001	2002000	2002001 Prime
4	10,000	20001	200020000	200020001
5	100,000	2<4>1	2<4>20<5>	2<4>2<4>1
6	1,000,000	2<5>1	2<5>2<6>	2<5>2<5>1 Prime
7	10**7	2<6>1	2<6>2<7>	2<6>2<6>1
8	10**8	2<7>1	2<7>2<8>	2<7>2<7>1
9	10**9	2<8>1	2<8>2<9>	2<8>2<8>1
10	10**10	2<9>1	2<9>2<10>	2<9>2<9>1 Prime
11	10**11	2<10>1	2<10>2<11>	2<10>2<10>1
12	10**12	2<11>1	2<11>2<12>	2<11>2<11>1
13	10**13	2<12>1	2<12>2<13>	2<12>2<12>1
14	10**14	2<13>1	2<13>2<14>	2<13>2<13>1
15	10**15	2<14>1	2<14>2<15>	2<14>2<14>1
16	10**16	2<15>1	2<15>2<16>	2<15>2<15>1
17	10**17	2<16>1	2<16>2<17>	2<16>2<16>1
18	10**18	2<17>1	2<17>2<18>	2<17>2<17>1
19	10**19	2<18>1	2<18>2<19>	2<18>2<18>1
20	10**20	2<19>1	2<19>2<20>	2<19>2<19>1
21	10**21	2<20>1	2<20>2<21>	2<20>2<20>1
22	10**22	2<21>1	2<21>2<22>	2<21>2<21>1
23	10**23	2<22>1	2<22>2<23>	2<22>2<22>1
24	10**24	2<23>1	2<23>2<24>	2<23>2<23>1
25	10**25	2<24>1	2<24>2<25>	2<24>2<24>1
26	10**26	2<25>1	2<25>2<26>	2<25>2<25>1
27	10**27	2<26>1	2<26>2<27>	2<26>2<26>1
28	10**28	2<27>1	2<27>2<28>	2<27>2<27>1
29	10**29	2<28>1	2<28>2<29>	2<28>2<28>1
30	10**30	2<29>1	2<29>2<30>	2<29>2<29>1
31	10**31	2<30>1	2<30>2<31>	2<30>2<30>1
32
33
34
710*6	10*(710**6)	---	----	2--(7 million zeros)--2--(7 million zeros)--1)
N large	10**(N large)	---	---	2--(N large zeros)--2--(N large zeros)--1

submitted to the Journal Nature

An Alternative to Mersenne Primes:

Finding the Largest Prime Using Convergent Sequences of Primitive Pythagorean Triples

2--(7 million zeroes)--2--(7 million zeros)--1

by Richard Allen Brown and Dr. John McKee

1248 Insititute, Charleston and Santa Barbara

Powered by Mathematica

Summary

Table 1. The first 10,000 Pythagorean Triples Coverging to an angle of ZERO

Table 2. The First 31 Primitive Pythagorean Triples of the Form n= 10**k, for all k = 1 through 31

Definition: 20000200001 = 2<4>2<4>1