Prime constant

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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)

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

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