Prime constant
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 numbers. That is,
where indicates a prime and
is the characteristic function of the primes.
The beginning of the decimal expansion of ρ is: (sequence A051006 in OEIS)
The beginning of the binary expansion is: (sequence A010051 in OEIS)
Irrationality
The number 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.
Notes
References
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