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Old 2020-09-15, 07:57   #1
sweety439
 
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Nov 2016

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Default Irregular primes and other types of primes

Primes related to Fermat's Last Theorem:

Bernoulli-irregular primes: 37, 59, 67, 101, 103, 131, 149, 157, 233, 257, 263, 271, 283, 293, 307, 311, 347, 353, 379, 389, 401, 409, 421, 433, 461, 463, 467, 491, 523, 541, 547, 557, 577, 587, 593, 607, 613, 617, 619, 631, 647, 653, 659, 673, 677, 683, 691, 727, 751, 757, 761, 773, 797, 809, 811, 821, 827, 839, 877, 881, 887, 929, 953, 971, 1061, ... [set A]

Euler-irregular primes: 19, 31, 43, 47, 61, 67, 71, 79, 101, 137, 139, 149, 193, 223, 241, 251, 263, 277, 307, 311, 349, 353, 359, 373, 379, 419, 433, 461, 463, 491, 509, 541, 563, 571, 577, 587, 619, 677, 691, 709, 739, 751, 761, 769, 773, 811, 821, 877, 887, 907, 929, 941, 967, 971, 983, 1013, 1019, ... [set B]

Primes such that k*p+1 is composite for all k in {2,4,8,10,14,16}: 197, 223, 227, 229, 257, 263, 283, 311, 317, 379, 383, 389, 457, 461, 463, 467, 521, 541, 569, 607, 661, 701, 751, 773, 787, 839, 859, 863, 881, 887, 907, 971, 991, ... [set C]

Consider the odd primes p, found the smallest n such that k*p^n+1 is prime for k in {2,4,8,10,14,16} (the primes in set C are the primes p such that these numbers are all composite for n=1): (italic if n>10000)

Code:
p      k=2       k=4       k=8     k=10     k=14      k=16
3: 1, 1, 2, 1, 1, 3,
5: 1, 2, 1, 2, 1, 2,
7: covering set, 1, covering set, 1, covering set, 1,
11: 1, covering set, 1, 10, covering set, 8,
13: covering set, 1, covering set, 1, covering set, 3,
17: 47, 6, 1, 1356, 1, 4,
19: covering set, 3, covering set, 1, covering set, 6, [E-irregular]
23: 1, 342, 119215, covering set, 5, 4,
29: 1, covering set, 1, 4, 3, 2,
31: covering set, covering set, covering set, 1, covering set, 2, [E-irregular]
37: covering set, 1, covering set, 2, covering set, 1, [B-irregular]
41: 1, covering set, covering set, 2, covering set, 4,
43: covering set, 1, covering set, 1, covering set, 3, [E-irregular]
47: 175, 2, covering set, 2, 1, covering set, [E-irregular]
53: 1, >1610000, 227183, 16, 1, 4
59: 3, covering set, 5, 36, 1, 2, [B-irregular]
61: covering set, covering set, covering set, 165, covering set, 1, [E-irregular]
67: covering set, 1, covering set, covering set, covering set, 3, [B-irregular] [E-irregular]
71: 3, covering set, 1, 2, covering set, 2, [E-irregular]
73: covering set, 1, covering set, 3, covering set, 40,
79: covering set, 1, covering set, 5, covering set, 8, [E-irregular]
83: 1, 5870, covering set, 2, 1, 348, 
89: 1, covering set, 5, covering set, 3, unknown
97: covering set, 1, covering set, 1, covering set, 1,
101: 192275, covering set, 1, 1506, covering set, covering set [B-irregular] [E-irregular]
103: covering set, 2, covering set, 1, covering set, covering set, [B-irregular]
107: 3, 32586, 1, 42, 1, 12
109: covering set, 3, covering set, 1, covering set, 2,
113: 1, 2958, 47, 2, 1, 40
127: covering set, 1, covering set, 4, covering set, 4,
131: 1, covering set, 1, covering set, covering set, 8, [B-irregular]
137: 327, 18, 1, 102, 93, covering set, [E-irregular]
139: covering set, 1, covering set, 18, covering set, 2, [E-irregular]
149: 3, covering set, 1, 2, 1, 18, [B-irregular] [E-irregular]
151: covering set, covering set, covering set, 1, covering set, 1,
157: covering set, 2, covering set, 1, covering set, 3,
163: covering set, 1, covering set, 3, covering set, 1,
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