Prime constant
Encyclopedia
The prime constant is the real number
whose 
th binary digit is 1 if 
is prime
and 0 if n is composite
or 1.
In other words,
is simply the number whose binary expansion
corresponds to the indicator function of the set of prime number
s. That is,
where
indicates a prime and 
is the characteristic function of the primes.
The beginning of the decimal expansion of ρ is:
The beginning of the binary expansion is:
is easily shown to be irrational
. To see why, suppose it were rational.
Denote the
th digit of the binary expansion of 
by 
. Then, since 
is assumed rational, there must exist 
, 
positive integers such that
for all 
and all 
.
Since there are an infinite number of primes, we may choose a prime
. By definition we see that 
. As noted, we have 
for all 
. Now consider the case 
. We have 
, since 
is composite because 
. Since 
we see that 
is irrational.
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
whose th binary digit is 1 if is primePrime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
and 0 if n is composite
Composite number
A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
or 1.
In other words,
is simply the number whose binary expansionBinary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
corresponds to the indicator function of the set of prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s. That is,
where
indicates a prime and is the characteristic function of the primes.The beginning of the decimal expansion of ρ is:
The beginning of the binary expansion is:
Irrationality
The number is easily shown to be irrationalIrrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
. To see why, suppose it were rational.
Denote the
th digit of the binary expansion of by . Then, since is assumed rational, there must exist , positive integers such that for all and all .Since there are an infinite number of primes, we may choose a prime
. By definition we see that . As noted, we have for all . Now consider the case . We have , since is composite because . Since we see that is irrational.