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)


A close approximation can be obtained using the golden ratio and the polygon circumscribing constant:[1]

The prime constant has been used in the calculation of value of the inverse fine-structure constant, together with traditional Pythagorean triangles:[1]


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.


  1. ^ a b Sherbon, Michael A. (2012), Fundamental nature of the fine-structure constant, doi:10.2139/ssrn.2380218 

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