prime constant

The number ρ defined by


is known as the prime constant. It is simply the number whose binary expansion corresponds to the characteristic functionMathworldPlanetmathPlanetmathPlanetmath of the set of prime numbersMathworldPlanetmath. That is, its nth binary digit is 1 if n is prime and 0 if n is composite.

The beginning of the decimal expansion of ρ is:


The number ρ is easily shown to be irrational. To see why, suppose it were rational. Denote the kth digit of the binary expansion of ρ by rk. Then, since ρ is assumed rational, there must exist N, k positive integers such that rn=rn+ik for all n>N and all i.

Since there are an infiniteMathworldPlanetmathPlanetmath number of primes, we may choose a prime p>N. By definition we see that rp=1. As noted, we have rp=rp+ik for all i. Now consider the case i=p. We have rp+ik=rp+pk=rp(k+1)=0, since p(k+1) is composite because k+12. Since rprp(k+1) we see that ρ is irrational.

The partial continued fractionsMathworldPlanetmath of the prime constant can be found

Title prime constant
Canonical name PrimeConstant
Date of creation 2013-03-22 15:02:17
Last modified on 2013-03-22 15:02:17
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 12
Author mathcam (2727)
Entry type Definition
Classification msc 11A41