The Copeland–Erdős constant is the concatenation of "0." with the
base 10 representations of the
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s in order. Its value, using the modern definition of prime, is approximately
:0.235711131719232931374143… .
The constant is
irrational
Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
; this can be proven with
Dirichlet's theorem on arithmetic progressions
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers ''a'' and ''d'', there are infinitely many primes of the form ''a'' + ''nd'', where ''n'' is al ...
or
Bertrand's postulate
In number theory, Bertrand's postulate is a theorem stating that for any integer n > 3, there always exists at least one prime number p with
:n < p < 2n - 2.
A less restrictive formulation is: for every , there is always ...
(Hardy and Wright, p. 113) or
Ramare's theorem that every
even integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
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 an
arithmetic progression
An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
''dn'' + ''a'', where ''a'' is
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
to ''d'' and to 10, will be irrational; for example, primes of the form 4''n'' + 1 or 8''n'' + 1. By Dirichlet's theorem, the arithmetic progression ''dn'' · 10
''m'' + ''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
In mathematics, a real number is said to be simply normal in an integer base b if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b. A number is said to ...
, a fact proven by
Arthur Herbert Copeland and
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...
in 1946 (hence the name of the constant).
The constant is given by
:
where ''p
n'' is the ''n''th
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
.
Its
continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
is
; 4, 4, 8, 16, 18, 5, 1, …().
Related constants
Copeland and Erdős's proof that their constant is normal relies only on the fact that
is
strictly increasing
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
and
, where
is the ''n''
th prime number. More generally, if
is any strictly increasing sequence of
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
s such that
and
is any natural number greater than or equal to 2, then the constant obtained by concatenating "0." with the
base-
representations of the
's is normal in base
. For example, the sequence
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 base ''b'' the number
:
which can be written in base ''b'' as 0.0110101000101000101…
''b''
where the ''n''th digit is 1 if and only if ''n'' is prime, is irrational.
See also
*
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.
References
Sources
*.
*.
External links
*
{{DEFAULTSORT:Copeland-Erdos constant
Paul Erdős
Irrational numbers
Prime numbers
Mathematical constants