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Constant Primes


Let p be a prime with n digits and let A be a constant. Call p an "A-prime" if the concatenation of the first n digits of A (ignoring the decimal point if one is present) give p. 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.

constantname of primescOEISn giving prime
Apéry's constantzeta(3)A11933410, 55, 109, 141
Catalan's constantCA11832852, 276, 25477
Champernowne constantCA07162010, 14, 24, 235, 2804, 4347, 37735, 68433
Copeland-Erdős constantCA2275301, 2, 4, 11, 353, 355, 499, 1171, 1543, 5719, 11048
ee-primeeA0641181, 3, 7, 85, 1781, 2780, 112280, 155025
Euler-Mascheroni constantgammaA0658151, 3, 40, 185, 1038, 22610, 179849
Glaisher-Kinkelin constantAA1184207, 10, 18, 64, 71, 527, 1992, 5644, 8813, 19692
Golomb-Dickman constantlambdaA1749746, 27, 57, 60, 1659, 2508
golden ratiophi-primephiA0641197, 13, 255, 280, 97241
Khinchin's constantKA1183271, 407, 878, 4443, 4981, 6551, 13386, 28433
natural logarithm of 2ln2A228226321, 466, 1271, 15690, 18872, 89973
natural logarithm of 10ln10A2282401, 2, 40, 242, 842, 1541, 75067
pipi-primepiA0604212, 6, 38, 16208, 47577, 78073, 613373
Pythagoras's constantsqrt(2)A11537755, 97, 225, 11260, 11540
Soldner's constantmuA1224224, 144, 227, 444, 19474
Theodorus's constantsqrt(3)A1193442, 3, 19, 111, 116, 641, 5411, 170657

The following table summarizes discoverers and discovery dates for some large constant primes.

constantdigitsdiscoverer
Apéry's constant19692E. W. Weisstein (Apr. 29, 2006)
Champernowne constant37735E. W. Weisstein (Jul. 15, 2013)
Copeland-Erdős constant11048E. W. Weisstein (Jul. 14, 2013)
Copeland-Erdős constant68433E. W. Weisstein (Aug. 16, 2013)
Copeland-Erdős constant97855E. W. Weisstein (Oct. 24, 2015)
Copeland-Erdős constant292447M. Rodenkirch (Dec. 11, 2015)
e112280E. W. Weisstein (Jul. 3, 2009)
e155025E. W. Weisstein (Oct. 7, 2010)
Euler-Mascheroni constant22610E. W. Weisstein (Apr. 25, 2006)
Euler-Mascheroni constant179849E. W. Weisstein (Jun. 1, 2011)
Khinchin's constant13386E. W. Weisstein (Apr. 26, 2006)
Khinchin's constant28433E. W. Weisstein (Apr. 27, 2006)
natural logarithm of 215690E. W. Weisstein (Aug. 17, 2013)
natural logarithm of 218872E. W. Weisstein (Aug. 18, 2013)
natural logarithm of 289973E. W. Weisstein (Oct. 28, 2015)
natural logarithm of 1075067E. W. Weisstein (Oct. 10, 2015)
pi47577E. W. Weisstein (Apr. 1, 2006)
pi16208E. W. Weisstein (Jan. 18, 2006)
pi78073E. W. Weisstein (Jul. 13, 2006)
pi613373A. Bondrescu (May 29, 2016)
golden ratio97289E. W. Weisstein (Jun. 4, 2009)
Pythagoras's constant11260E. W. Weisstein (Jan. 21, 2006)
Pythagoras's constant11540E. W. Weisstein (Jan. 21, 2006)
Theodorus's constant170657E. W. Weisstein (Aug. 18, 2013)

The following table summarizes the values of known constant primes for some named mathematical constants. The first of the sqrt(2)-primes (where sqrt(2) 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.

constantcOEISprimes
Apéry's constantzeta(3)A1193331202056903, 1202056903159594285399738161511449990764986292340498881, ...
Champernowne constantCA1769421234567891, 12345678910111, 123456789101112131415161, ...
Catalan's constantCA1183299159655941772190150546035149323841107741493742816721, ...
Copeland-Erdős constantCA2275292, 23, 2357, 23571113171, ...,
eeA0075122, 271, 2718281, ...
Euler-Mascheroni constantgammaA0729525, 577, 5772156649015328606065120900824024310421, ...
Glaisher-Kinkelin constantAA1184191282427, 1282427129, 128242712910062263, ...
Golomb-Dickman constantlambdaA174975624329, 624329988543550870992936383, ...
golden ratiophiA0641171618033, 1618033988749, ...
natural logarithm of 10ln10A2282412, 23, 2302585092994045684017991454684364207601, ...
pipiA0050423, 31, 314159, 31415926535897932384626433832795028841, ...
Pythagoras's constantsqrt(2)A1154531414213562373095048801688724209698078569671875376948073, ...
Soldner's constantmuA1224221451, ...
Theodorus's constantsqrt(3)A11934317, 173, 1732050807568877293, ...

See also

Consecutive Number Sequences, Constant Digit Scanning, e-Prime, Integer Sequence Primes, Phi-Prime, Pi-Prime, Prime Constant

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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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Constant Primes

Cite this as:

Weisstein, Eric W. "Constant Primes." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ConstantPrimes.html

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