Copeland–Erdős constant: Difference between revisions
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The constant is irrational. By [[Dirichlet's theorem on arithmetic progressions]], for any ''m'', there exist primes of the form |
The constant is irrational. By [[Dirichlet's theorem on arithmetic progressions]], for any ''m'', there exist primes of the form |
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:<math>k 10^{m+1} + 1 .</math> |
:<math>k 10^{m+1} + 1 .</math> |
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Hence, there exist primes with digit strings containing at least ''m'' zeros followed by the digit 1. Thus, the digit string of the Copeland-Erdős constant |
Hence, there exist primes with digit strings containing at least ''m'' 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). |
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By a similar argument, any constant created by concatenating "0." with all primes in an [[arithmetic progression]] <math>d \cdot n + a</math>, where ''a'' is [[coprime]] to ''d'' and to 10, |
By a similar argument, any constant created by concatenating "0." with all primes in an [[arithmetic progression]] <math>d \cdot n + a</math>, where ''a'' is [[coprime]] to ''d'' and to 10, will be irrational. E.g. primes of the form <math>4n+1</math> or <math>8n-1</math>. By Dirichlet's theorem, the arithmetic progression <math>d \cdot n \cdot 10^m + a</math> contains primes for all ''m'', and those primes are also in <math>d \cdot n + a</math>, so the concatenated primes contain arbitrarily long sequences of the digit zero. |
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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). |
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). |
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Revision as of 23:13, 13 August 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, for any m, there exist primes of the form
Hence, there exist primes with digit strings containing at least m 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).
By a similar argument, any constant created by concatenating "0." with all primes in an arithmetic progression , where a is coprime to d and to 10, will be irrational. E.g. primes of the form or . By Dirichlet's theorem, the arithmetic progression contains primes for all m, and those primes are also in , so the concatenated primes contain arbitrarily long sequences of the digit zero.
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.