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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 the OEIS)

The beginning of the binary expansion is: (sequence A010051 in the OEIS)

IrrationalityEdit

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.

External linksEdit

  • Weisstein, Eric W. "Prime Constant". MathWorld.