Copeland–Erdős constant: Difference between revisions

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:0.235711131719232931374143... {{OEIS|id=A33308}}
:0.235711131719232931374143... {{OEIS|id=A33308}}


The constant is irrational. By [[Dirichlet's theorem on arithmetic progressions]], there exist primes of the form
⚫
It is given by
:<math>k 10^m + 1</math>
for all positive integers <math>k</math> and <math>m</math>. 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).


By a simlar argument, any constant created by concatenating all primes in a given arithmetic progression (e.g., primes of the form <math>4n+1</math> or <math>8n-1</math>) will be irrational.
⚫
:<math>\sum_{n=1}^\infty p(n) 10^{-(n + \sum_{k=1}^n floor(\log_{10}{p(n)}))}</math>


⚫
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
⚫
:<math>\displaystyle \sum_{n=1}^\infty p(n) 10^{-\left(n + \sum_{k=1}^n \lfloor \log_{10}{p(n)} \rfloor \right)}</math>
where p(n) gives the n-th [[prime number]].
where p(n) gives the n-th [[prime number]].

⚫
The larger [[Smarandache-Wellin number]]s approximate the value of this constant multiplied by the appropriate power of 10.


Its [[continued fraction]] is [0; 4, 4, 8, 16, 18, 5, 1, ...] ({{OEIS2C|id=A30168}})
Its [[continued fraction]] is [0; 4, 4, 8, 16, 18, 5, 1, ...] ({{OEIS2C|id=A30168}})


⚫
The larger [[Smarandache-Wellin number]]s approximate the value of this constant multiplied by the appropriate power of 10.
⚫
In base 10, this is a [[normal number]], a fact proven by [[Arthur Herbert Copeland]] and [[Paul Erdős]] in 1946 (hence the name of the constant).

The Copeland-Erdős constant is irrational. By the theorem called [[Bertrand's postulate]], there are at least 3 primes of ''n'' digits for every ''n''; in fact, there are many more. Let us suppose that the constant were rational, say ''k'', and suppose the first prime of 2''k'' digits lies in the [[periodic]] part of the rational number. But then that prime consists of two complete repetitions of a transform of the period, or of two identical sequences of digits concatenated together. but then it has 10<sup>k</sup>+1 as a factor, and is not prime.

(If the first prime of length 2''k'' fails the necessary condition, consider the first prime of length 3''k''. 4''k'' etc. One of them must; the non-periodic part of a rational can only have finite length.)

The same thing is true of the numbers formed by concatenating all primes of the form 4''n''+1 and all primes of the form 4''n''-1; the proof is a trivial variation of the above.


==References==
==References==


* [[G. H. Hardy|Hardy G. H.]] and [[E. M. Wright]] ([[1938]]) ''An Introduction to the Theory of Numbers'', Oxford University Press, USA; 5 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.




Revision as of 18:54, 17 May 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

0.235711131719232931374143... (sequence A33308 in the OEIS)

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).

By a simlar argument, any constant created by concatenating all primes in a given arithmetic progression (e.g., primes of the form or ) will be irrational.

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.


allikas: https://en.wikipedia.org/wiki/Special%3ADiff%2F131598317