The prime constant is the real number whose
thbinary digit is 1 if
isprime and 0 if
iscomposite or 1.
In other words, is the number whose binary expansion corresponds to the indicator function of the setofprime numbers. That is,
where indicates a prime and
is the characteristic function of the set
of prime numbers.
The beginning of the decimal expansion of ρ is: (sequence A051006 in the OEIS)
The beginning of the binary expansion is: (sequence A010051 in the OEIS)
The number can be shown to be irrational.[1] To see why, suppose it were rational.
Denote the th digit of the binary expansion of
by
. Then since
is assumed rational, its binary expansion is eventually periodic, and so there exist positive integers
and
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.
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