Copeland–Erdős constant: Difference between revisions
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The larger [[Smarandache-Wellin number]]s approximate the value of this constant multiplied by the appropriate power of 10. |
The larger [[Smarandache-Wellin number]]s approximate the value of this constant multiplied by the appropriate power of 10. |
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== Analogue== |
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If the constant created by concatenating all primes of the form <math>4n+1</math> is rational, there exists an integer <math>s</math> such that there is no prime of the form <math>4n+1</math> of <math>s</math> digits. (Hardy and Wright, p. 113) |
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:0.5131729374153... |
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==References== |
==References== |
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* [[G. H. Hardy|Hardy G. H.]] and [[E. M. Wright]] ([[1938]]) ''An Introduction to the Theory of Numbers'', Oxford University Press, USA; 5th edition (April 17, 1980). ISBN 0-19-853171-0. |
* [[G. H. Hardy|Hardy G. H.]] and [[E. M. Wright]] ([[1938]]) ''An Introduction to the Theory of Numbers'', Oxford University Press, USA; 5th edition (April 17, 1980). ISBN 0-19-853171-0. |
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*{{MathWorld|title=Copeland-Erdos Constant|urlname=Copeland-ErdosConstant}} |
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Revision as of 11:36, 6 July 2007
The Copeland-Erdős constant is the concatenation of "0." with the base 10 representations of the prime numbers in order. Its value is approximately
The constant is irrational. By Dirichlet's theorem on arithmetic progressions, there exist primes of the form
for all positive integers and . Hence, there exist primes with digit strings containing arbitrarily long sequences of zeros followed by the digit 1. Thus, the digit string of the Copeland-Erdős constant contains arbitrarily long sequences of zeros followed by the digit 1, and hence the digit string of the constant cannot terminate or recur. So, the constant is irrational (Hardy and Wright, p. 113).
In base 10, the constant is a normal number, a fact proven by Arthur Herbert Copeland and Paul Erdős in 1946 (hence the name of the constant).
The constant is given by
where p(n) gives the n-th prime number.
Its continued fraction is [0; 4, 4, 8, 16, 18, 5, 1, ...] (OEIS: A30168)
The larger Smarandache-Wellin numbers approximate the value of this constant multiplied by the appropriate power of 10.
References
- Hardy G. H. and E. M. Wright (1938) An Introduction to the Theory of Numbers, Oxford University Press, USA; 5th edition (April 17, 1980). ISBN 0-19-853171-0.
- Weisstein, Eric W. "Copeland-Erdos Constant". MathWorld.