Copeland–Erdős constant: Difference between revisions
re-write section for greater clarity |
Michael Hardy (talk | contribs) copy edits per WP:MOS and WP:MOSMATH |
||
| Line 5: | Line 5: | ||
The constant can be proved to be irrational by the use of either [[Dirichlet's theorem on arithmetic progressions]] or [[Chebyshev's theorem]]. (Hardy and Wright, p. 113). |
The constant can be proved to be irrational by the use of either [[Dirichlet's theorem on arithmetic progressions]] or [[Chebyshev's theorem]]. (Hardy and Wright, p. 113). |
||
By a similar argument, any constant created by concatenating "0." with all primes in an [[arithmetic progression]] |
By a similar argument, any constant created by concatenating "0." with all primes in an [[arithmetic progression]] ''dn'' + ''a'', where ''a'' is [[coprime]] to ''d'' and to 10, will be irrational. E.g. primes of the form 4''n'' + 1</math> or 8''n'' + 1. By Dirichlet's theorem, the arithmetic progression ''dn·10<sup>''m''</sup> + ''a'' contains primes for all ''m'', and those primes are also in ''cd'' + ''a'', 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). |
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). |
||
| Line 11: | Line 11: | ||
The constant is given by |
The constant is given by |
||
:<math>\displaystyle \sum_{n=1}^\infty p(n) 10^{-\left(n + \sum_{k=1}^n \lfloor \log_{10}{p(k)} \rfloor \right)}</math> |
:<math>\displaystyle \sum_{n=1}^\infty p(n) 10^{-\left(n + \sum_{k=1}^n \lfloor \log_{10}{p(k)} \rfloor \right)}</math> |
||
where p(n) |
where ''p''(''n'') is the ''n''th [[prime number]]. |
||
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}}). |
||
| Line 19: | Line 20: | ||
In any given base ''b'' the number |
In any given base ''b'' the number |
||
: |
: <math>\displaystyle \sum_{n=1}^\infty b^{-p(n)}, \, </math> |
||
which can be written in base ''b'' as 0.0110101000101000101…<sub>''b''</sub> |
which can be written in base ''b'' as 0.0110101000101000101…<sub>''b''</sub> |
||
where the ''n'' |
where the ''n''th digit is 1 if ''n'' is prime, is irrational. (Hardy and Wright, p. 112). |
||
==See also== |
==See also== |
||
*[[ |
*[[Smarandache–Wellin number]]s: the truncated value of this constant multiplied by the appropriate power of 10. |
||
==References== |
==References== |
||
Revision as of 16:22, 14 June 2009
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 can be proved to be irrational by the use of either Dirichlet's theorem on arithmetic progressions or Chebyshev's theorem. (Hardy and Wright, p. 113).
By a similar argument, any constant created by concatenating "0." with all primes in an arithmetic progression dn + a, where a is coprime to d and to 10, will be irrational. E.g. primes of the form 4n + 1</math> or 8n + 1. By Dirichlet's theorem, the arithmetic progression dn·10m + a contains primes for all m, and those primes are also in cd + a, 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) is the nth prime number.
Its continued fraction is [0; 4, 4, 8, 16, 18, 5, 1, …] (OEIS: A30168).
Related constants
In any given base b the number
which can be written in base b as 0.0110101000101000101…b where the nth digit is 1 if n is prime, is irrational. (Hardy and Wright, p. 112).
See also
- Smarandache–Wellin numbers: the truncated value of this constant multiplied by the appropriate power of 10.
References
- Hardy, G. H.; Wright, E. M. (1980), An Introduction to the Theory of Numbers (5th ed.), Oxford University Press, ISBN 0-19-853171-0.