Permanent Copeland - Erdos

The Copeland – Erdös constant is a real number constructed as the concatenation “0,” (“zero integers ...”) with a concatenated sequence of increasing primes in decimal notation ^[1] :

0.235711131719232931374143 ...

The constant is irrational ; this fact can be proved using the Dirichlet theorem on primes in arithmetic progression or Bertrand’s postulate ^[2] or Ramare’s theorem (which states that any even integer is the sum of at most six primes). This fact also follows from the fact that this constant is a normal number ; the normality of a constant in decimal notation was proved in 1949 by Arthur Heropel Copeland and Pal Erdös .

Any constant formed by the concatenation "0," with all primes in arithmetic progression $dn+a$ ${\ displaystyle dn + a}$ ${\ displaystyle dn + a}$ where $a$ ${\ displaystyle a}$ $a$ - a mutually prime number with a number $d$ ${\ displaystyle d}$ $d$ and the number 10 will be irrational. For example, these are prime numbers taking the form $4n+1$ ${\ displaystyle 4n + 1}$ ${\ displaystyle 4n + 1}$ or $8n+1$ ${\ displaystyle 8n + 1}$ ${\ displaystyle 8n + 1}$ . According to Dirichlet's theorem, arithmetic progression $dn\cdot 10^{m}+a$ ${\ displaystyle dn \ cdot 10 ^ {m} + a}$ ${\ displaystyle dn \ cdot 10 ^ {m} + a}$ contains prime numbers for any number $m$ ${\ displaystyle m}$ $m$ , and these primes are also in $cd+a$ ${\ displaystyle cd + a}$ ${\ displaystyle cd + a}$ therefore, among these concatenated primes, any desired number of zeros following each other will be contained.

Copeland's constant - Erdös can be expressed as:

\displaystyle \sum _{n=1}^{\infty }p_{n}10^{-\left(n+\sum _{k=1}^{n}\lfloor \log _{10}{p_{k}}\rfloor \right)}

{\ displaystyle \ displaystyle \ sum _ {n = 1} ^ {\ infty} p_ {n} 10 ^ {- \ left (n + \ sum _ {k = 1} ^ {n} \ lfloor \ log _ {10} {p_ {k}} \ rfloor \ right)}}

{\ displaystyle \ displaystyle \ sum _ {n = 1} ^ {\ infty} p_ {n} 10 ^ {- \ left (n + \ sum _ {k = 1} ^ {n} \ lfloor \ log _ {10} {p_ {k}} \ rfloor \ right)}}

,

Where $p_{n}$ ${\ displaystyle p_ {n}}$ $p_n$ - this $n$ ${\ displaystyle n}$ $n$ is a prime number .

Continuous fraction of a number - [0; 4, 4, 8, 16, 18, 5, 1, ...] ^[3] .

Similar Constants

For any positional number system with a base $b$ ${\ displaystyle b}$ number:

\displaystyle \sum _{n=1}^{\infty }b^{-p_{n}}

{\ displaystyle \ displaystyle \ sum _ {n = 1} ^ {\ infty} b ^ {- p_ {n}}}

,

which can be written in this number system as 0.0110101000101000101 ..., where $n$ ${\ displaystyle n}$ the 1st digit is 1 if $n$ ${\ displaystyle n}$ Is a prime, is irrational ^[4] .

Chemternoun's constant is the concatenation of all positive integers, not just primes.

Notes

↑ sequence A033308 in OEIS
↑ Hardy, Wright, 1938 .
↑ A030168
↑ Hardy, Wright, 1938 , p. 112.

Links

Weisstein, Eric W. Copeland-Erdos Constant on Wolfram MathWorld .
GH Hardy , EM Wright. An Introduction to the Theory of Numbers. - 5 ^{th ed.} . - Oxford University Press , 1938. - ISBN 0-19-853171-0 .

[1] sequence A033308 in OEIS

[_3ddd39ec3edbb86c-2] Hardy, Wright, 1938 .

[3] A030168

[_78f4d75117c122e8-4] Hardy, Wright, 1938 , p. 112.

Permanent Copeland - Erdos

Similar Constants

Notes

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