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علماء الرياضيات | Constant Primes |
 638
 02:15 صباحاً
date: 21-9-2020 |
Author : Pickover, C. A. 
Book or Source : The Mathematics of Oz: Mental Gymnastics from Beyond the Edge. New York: Cambridge University Press, 2002. 
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Date: 25-11-2020
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Let 
be a prime with 
digits and let 
be a constant. Call 
an "
-prime" if the concatenation of the first 
digits of 
(ignoring the decimal point if one is present) give 
. Constant primes are therefore a special type of integer sequence primes, with e-primes, pi-primes, and phi-primes being perhaps the most prominent examples.
The following table summarizes the indices of known constant primes for some named mathematical constants.
| constant | name of primes |  |
OEIS |  giving prime |
| Apéry's constant |  |
A119334 | 10, 55, 109, 141 | |
| Catalan's constant |  |
A118328 | 52, 276, 25477 | |
| Champernowne constant |  |
A071620 | 10, 14, 24, 235, 2804, 4347, 37735, 68433 | |
| Copeland-Erdős constant |  |
A227530 | 1, 2, 4, 11, 353, 355, 499, 1171, 1543, 5719, 11048 | |
| e | e-prime |  |
A064118 | 1, 3, 7, 85, 1781, 2780, 112280, 155025 |
| Euler-Mascheroni constant |  |
A065815 | 1, 3, 40, 185, 1038, 22610, 179849 | |
| Glaisher-Kinkelin constant |  |
A118420 | 7, 10, 18, 64, 71, 527, 1992, 5644, 8813, 19692 | |
| Golomb-Dickman constant |  |
A174974 | 6, 27, 57, 60, 1659, 2508 | |
| golden ratio | phi-prime |  |
A064119 | 7, 13, 255, 280, 97241 |
| Khinchin's constant |  |
A118327 | 1, 407, 878, 4443, 4981, 6551, 13386, 28433 | |
| natural logarithm of 2 |  |
A228226 | 321, 466, 1271, 15690, 18872, 89973 | |
| natural logarithm of 10 |  |
A228240 | 1, 2, 40, 242, 842, 1541, 75067 | |
| pi | pi-prime |  |
A060421 | 2, 6, 38, 16208, 47577, 78073, 613373 |
| Pythagoras's constant |  |
A115377 | 55, 97, 225, 11260, 11540 | |
| Soldner's constant |  |
A122422 | 4, 144, 227, 444, 19474 | |
| Theodorus's constant |  |
A119344 | 2, 3, 19, 111, 116, 641, 5411, 170657 |
The following table summarizes discoverers and discovery dates for some large constant primes.
| constant | digits | discoverer |
| Apéry's constant | 19692 | E. W. Weisstein (Apr. 29, 2006) |
| Champernowne constant | 37735 | E. W. Weisstein (Jul. 15, 2013) |
| Copeland-Erdős constant | 11048 | E. W. Weisstein (Jul. 14, 2013) |
| Copeland-Erdős constant | 68433 | E. W. Weisstein (Aug. 16, 2013) |
| Copeland-Erdős constant | 97855 | E. W. Weisstein (Oct. 24, 2015) |
| Copeland-Erdős constant | 292447 | M. Rodenkirch (Dec. 11, 2015) |
| e | 112280 | E. W. Weisstein (Jul. 3, 2009) |
| e | 155025 | E. W. Weisstein (Oct. 7, 2010) |
| Euler-Mascheroni constant | 22610 | E. W. Weisstein (Apr. 25, 2006) |
| Euler-Mascheroni constant | 179849 | E. W. Weisstein (Jun. 1, 2011) |
| Khinchin's constant | 13386 | E. W. Weisstein (Apr. 26, 2006) |
| Khinchin's constant | 28433 | E. W. Weisstein (Apr. 27, 2006) |
| natural logarithm of 2 | 15690 | E. W. Weisstein (Aug. 17, 2013) |
| natural logarithm of 2 | 18872 | E. W. Weisstein (Aug. 18, 2013) |
| natural logarithm of 2 | 89973 | E. W. Weisstein (Oct. 28, 2015) |
| natural logarithm of 10 | 75067 | E. W. Weisstein (Oct. 10, 2015) |
| pi | 47577 | E. W. Weisstein (Apr. 1, 2006) |
| pi | 16208 | E. W. Weisstein (Jan. 18, 2006) |
| pi | 78073 | E. W. Weisstein (Jul. 13, 2006) |
| pi | 613373 | A. Bondrescu (May 29, 2016) |
| golden ratio | 97289 | E. W. Weisstein (Jun. 4, 2009) |
| Pythagoras's constant | 11260 | E. W. Weisstein (Jan. 21, 2006) |
| Pythagoras's constant | 11540 | E. W. Weisstein (Jan. 21, 2006) |
| Theodorus's constant | 170657 | E. W. Weisstein (Aug. 18, 2013) |
The following table summarizes the values of known constant primes for some named mathematical constants. The first of the 
-primes (where 
is Pythagoras's constant) was found by J. Earls (Pickover 2002, p. 334) and, contrary to Pickover's claim, is actually the smallest (rather than the largest known) example.
| constant |  |
OEIS | primes |
| Apéry's constant |  |
A119333 | 1202056903, 1202056903159594285399738161511449990764986292340498881, ... |
| Champernowne constant |  |
A176942 | 1234567891, 12345678910111, 123456789101112131415161, ... |
| Catalan's constant |  |
A118329 | 9159655941772190150546035149323841107741493742816721, ... |
| Copeland-Erdős constant |  |
A227529 | 2, 23, 2357, 23571113171, ..., |
| e |  |
A007512 | 2, 271, 2718281, ... |
| Euler-Mascheroni constant |  |
A072952 | 5, 577, 5772156649015328606065120900824024310421, ... |
| Glaisher-Kinkelin constant |  |
A118419 | 1282427, 1282427129, 128242712910062263, ... |
| Golomb-Dickman constant |  |
A174975 | 624329, 624329988543550870992936383, ... |
| golden ratio |  |
A064117 | 1618033, 1618033988749, ... |
| natural logarithm of 10 |  |
A228241 | 2, 23, 2302585092994045684017991454684364207601, ... |
| pi |  |
A005042 | 3, 31, 314159, 31415926535897932384626433832795028841, ... |
| Pythagoras's constant |  |
A115453 | 1414213562373095048801688724209698078569671875376948073, ... |
| Soldner's constant |  |
A122422 | 1451, ... |
| Theodorus's constant |  |
A119343 | 17, 173, 1732050807568877293, ... |
REFERENCES:
Pickover, C. A. The Mathematics of Oz: Mental Gymnastics from Beyond the Edge. New York: Cambridge University Press, 2002.
Sloane, N. J. A. Sequences A005042, A007512, A060421, A064117, A064118, A064119, A065815, A071620, A072952, A115377, A115453, A118327, A118328, A118329, A118419, A118420, A119333, A119334, A119343, A119344, A122421, A122422, A174974, A174975, A176942, A227529, A227530, A228226, A228240, and A228241 in "The On-Line Encyclopedia of Integer Sequences."
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