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User:M. F. Hasler/Work in progress/Twin primes
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This is about some recent work on twin primes, mainly inspired by previous work by A. Dinculescu ([[#References|]] [D12]-[D18]) and personal communications with the author. The "Holy Grail" is the Twin prime conjecture, but for the moment we just try to establish some useful intermediate results.
Contents
- 1 Related sequences
- 1.1 A002822 Numbers n such that 6n-1, 6n+1 are twin primes.
- 1.2 A057767 Number of twin prime pairs between P(n)^2 and P(n+1)^2 where P(n) is the n-th prime.
- 1.3 A263282 Numbers n such that 6n is in A002822 but n is not.
- 1.4 Twin Goldbach conjecture related
- 1.5 Perfect power twin ranks related
- 1.5.1 A326229 Square array T(n,k) where row n >= 1 lists numbers m > 1 such that 6*m^n +- 1 are twin primes; read by falling antidiagonals.
- 1.5.2 A326230 Least k > 1 such that k^n is a twin rank (cf. A002822: 6*k^n +- 1 are twin primes).
- 1.5.3 A326231 Numbers n such that N = (5n)^2 is a twin rank (cf. A002822: 6N +- 1 are twin primes).
- 1.5.4 A326232 Numbers n such that N = n^2 is a twin rank (cf. A002822: 6N +- 1 are twin primes).
- 1.5.5 A326233 Numbers n such that N = (7n)^3 is a twin rank (A002822: 6N +- 1 are twin primes).
- 1.5.6 A326234 Numbers n such that N = n^3 is a twin rank (A002822: 6N +- 1 are twin primes).
- 1.5.7 A326235 Numbers n such that N = (35n)^6 is a twin rank (A002822: 6N +- 1 are twin primes).
- 1.5.8 A326236 Numbers n such that N = n^6 is a twin rank (cf. A002822: 6N +- 1 are twin primes).
- 1.6 Others
- 2 References
Related sequences
A002822 Numbers n such that 6n-1, 6n+1 are twin primes.
- 1, 2, 3, 5, 7, 10, 12, 17, 18, 23, 25, 30, 32, 33, 38, 40, 45, 47, 52, 58, 70, 72, 77, 87, 95, 100, 103, 107, 110, 135, 137, 138, 143, ...
A057767 Number of twin prime pairs between P(n)^2 and P(n+1)^2 where P(n) is the n-th prime.
- 1, 2, 2, 4, 2, 7, 2, 4, 8, 2, 11, 7, 3, 11, 13, 13, 5, 19, 11, 3, 15, 14, 14, 21, 15, 7, 10, 6, 11, 42, 12, 27, 6, 45, 10, 20, 17, 21,...
A263282 Numbers n such that 6n is in A002822 but n is not.
- 63, 65, 88, 98, 102, 133, 157, 163, 185, 193, 198, 203, 208, 210, 233, 245, 250, 262, 310, 340, 380, 387, 413, 437, 457, 462, 473, 478,...
A326228 Primes p such that the distance from p#/6 to the next larger or smaller twin rank is not in A002822. (# = primorial, A034386)
- 41, 227, 307, 311, 349, 457, 613
A326229 Square array T(n,k) where row n >= 1 lists numbers m > 1 such that 6*m^n +- 1 are twin primes; read by falling antidiagonals.
- 2, 3, 5, 5, 10, 28, 7, 35, 42, 70, 10, 60, 168, 75, 2, 12, 70, 203, 80, 40, 1820, 17, 75, 287, 175, 208, 2590, 110, 18, 210, 308, 485...
A326230 Least k > 1 such that k^n is a twin rank (cf. A002822: 6*k^n +- 1 are twin primes).
- 2, 5, 28, 70, 2, 1820, 110, 1850, 2520, 220, 2023, 9415, 647, 2880, 2562, 3895, 2, 51240, 525, 3750, 147, 2350, 355, 4480, 2588, 3370,...
A326231 Numbers n such that N = (5n)^2 is a twin rank (cf. A002822: 6N +- 1 are twin primes).
- 1, 2, 7, 12, 14, 15, 42, 48, 77, 86, 89, 99, 118, 131, 146, 161, 163, 167, 201, 208, 209, 246, 278, 286, 306, 334, 343, 370, 378, 384,...
A326232 Numbers n such that N = n^2 is a twin rank (cf. A002822: 6N +- 1 are twin primes).
- 1, 5, 10, 35, 60, 70, 75, 210, 240, 385, 430, 445, 495, 590, 655, 730, 805, 815, 835, 1005, 1040, 1045, 1230, 1390, 1430, 1530, 1670,...
A326233 Numbers n such that N = (7n)^3 is a twin rank (A002822: 6N +- 1 are twin primes).
- 4, 6, 24, 29, 41, 44, 74, 149, 151, 216, 229, 234, 240, 251, 284, 415, 481, 561, 574, 704, 719, 735, 751, 756, 776, 819, 966, 1026, ...
A326234 Numbers n such that N = n^3 is a twin rank (A002822: 6N +- 1 are twin primes).
- 1, 28, 42, 168, 203, 287, 308, 518, 1043, 1057, 1512, 1603, 1638, 1680, 1757, 1988, 2905, 3367, 3927, 4018, 4928, 5033, 5145, 5257, ...
A326235 Numbers n such that N = (35n)^6 is a twin rank (A002822: 6N +- 1 are twin primes).
- 52, 74, 137, 159, 238, 242, 304, 306, 456, 478, 547, 701, 756, 988, 1059, 1186, 1218, 1243, 1378, 1705, 1976, 2426, 2596, 2844, 2952, ...
A326236 Numbers n such that N = n^6 is a twin rank (cf. A002822: 6N +- 1 are twin primes).
- 1, 1820, 2590, 4795, 5565, 8330, 8470, 10640, 10710, 15960, 16730, 19145, 24535, 26460, 34580, 37065, 41510, 42630, 43505, 48230, ...
Others
A308344(n) = (A001359(n+1)^2 - 1)/24, where A001359 = lesser of twin primes
- 1, 5, 12, 35, 70, 145, 210, 425, 477, 782, 925, 1335, 1520, 1617, 2147, 2380, 3015, 3290, 4030, 5017, 7315, 7740, 8855, 11310, 13490, 14950, 15862, 17120, 18095, 27270, 28085, 28497, 30602, 32340, 43265, 44290, 45850, 46905, 49595, 55200, 62935, 67947, 69230, 7052
References
- [D12] A. Dinculescu: On Some Infinite Series Related to the Twin Primes.
- The Open Mathematics Journal, 5 (2012), 8--14. doi: 10.2174/1874117701205010008
- [D13] A. Dinculescu: The Twin Primes Seen from a Different Perspective.
- British Journal of Mathematics & Computer Science, 3 no 4 (2013) 691--698. doi: 10.9734/BJMCS/2013/4358
- [D16] A. Dinculescu: A Twin Prime Analog of Goldbach’s Conjecture.
- British Journal of Mathematics & Computer Science, 12 no 5 (2016) 1--4. doi: 10.9734/BJMCS/2016/22064 - Article no.BJMCS.22064 - ISSN: 2231-0851
- [D18] A. Dinculescu: On the Numbers that Determine the Distribution of Twin Primes.
- Surveys in Mathematics and its Applications, 13 (2018), 171-181. URL: http://www.utgjiu.ro/math/sma/v13/a13_11.html