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 | ||
| 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, ... |