Constant Primes

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, ...

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