TheCopeland–Erdős constant is the concatenation of "0." with thebase 10 representations of theprime numbers in order. Its value, using the modern definition of prime,^[1] is approximately

0.235711131719232931374143... (sequenceA033308 in theOEIS).

The constant isirrational; this can be proven withDirichlet's theorem on arithmetic progressions orBertrand's postulate (Hardy and Wright, p. 113) orRamare's theorem that everyeveninteger is a sum of at most six primes. It also follows directly from its normality (see below).

By a similar argument, any constant created by concatenating "0." with all primes in anarithmetic progressiondn + a, wherea iscoprime tod and to 10, will be irrational; for example, primes of the form 4n + 1 or 8n + 1. By Dirichlet's theorem, the arithmetic progressiondn · 10^m + a contains primes for allm, and those primes are also incd + a, so the concatenated primes contain arbitrarily long sequences of the digit zero.

In base 10, the constant is anormal number, a fact proven byArthur Herbert Copeland andPaul Erdős in 1946 (hence the name of the constant).^[2]

The constant is given by

${\displaystyle {\displaystyle \sum _{n=1}^{\mathrm{\infty }}{p}_n{10}^{-\left(n+\sum _{k=1}^n\lfloor {\log }_{10}{p}_k\rfloor \right)}}}$

wherep_n is thenthprime number.

Itssimple continued fraction is [0; 4, 4, 8, 16, 18, 5, 1, ...] (OEIS: A030168).

Copeland and Erdős's proof that their constant is normal relies only on the fact that${\displaystyle p_n}$ isstrictly increasing and${\displaystyle p_n=n^{1+o(1)}}$, where${\displaystyle p_n}$ is then^th prime number. More generally, if${\displaystyle s_n}$ is any strictly increasing sequence ofnatural numbers such that${\displaystyle s_n=n^{1+o(1)}}$ and${\displaystyle b}$ is any natural number greater than or equal to 2, then the constant obtained by concatenating "0." with thebase-${\displaystyle b}$ representations of the${\displaystyle s_n}$'s is normal in base${\displaystyle b}$. For example, the sequence${\displaystyle \lfloor n(\log n{)}^2\rfloor }$ satisfies these conditions, so the constant 0.003712192634435363748597110122136... is normal in base 10, and 0.003101525354661104...₇ is normal in base 7.

In any given baseb the number

${\displaystyle {\displaystyle \sum _{n=1}^{\mathrm{\infty }}{b}^{-p_n},}}$

which can be written in baseb as 0.0110101000101000101..._bwhere thenth digit is 1 if and only ifn is prime, is irrational.^[3]

• Smarandache–Wellin numbers: the truncated value of this constant multiplied by the appropriate power of 10.
• Champernowne constant: concatenating all natural numbers, not just primes.

• Copeland, A. H.;Erdős, P. (1946), "Note on Normal Numbers",Bulletin of the American Mathematical Society,52 (10):857–860,doi:10.1090/S0002-9904-1946-08657-7.
• Hardy, G. H.;Wright, E. M. (1979) [1938],An Introduction to the Theory of Numbers (5th ed.), Oxford University Press,ISBN 0-19-853171-0.

• Weisstein, Eric W."Copeland-Erdos Constant".MathWorld.

• Paul Erdős
• Irrational numbers
• Prime numbers
• Mathematical constants

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• Short description is different from Wikidata