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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)


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.