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Prime constant

Real number whose nth binary digit is 1 if n is prime and 0 if n is composite or 1

Summary

Real number whose nth binary digit is 1 if n is prime and 0 if n is composite or 1

The prime constant is the real number \rho whose nth binary digit is 1 if n is prime and 0 if n is composite or 1.

In other words, \rho is the number whose binary expansion corresponds to the indicator function of the set of prime numbers. That is, : \rho = \sum_{p} \frac{1}{2^p} = \sum_{n=1}^\infty \frac{\chi_{\mathbb{P}}(n)}{2^n} where p indicates a prime and \chi_{\mathbb{P}} is the characteristic function of the set \mathbb{P} of prime numbers.

The beginning of the decimal expansion of ρ is: \rho = 0.414682509851111660248109622\ldots

The beginning of the binary expansion is: \rho = 0.011010100010100010100010000\ldots_2

Irrationality

The number \rho is irrational.

Proof by contradiction

Suppose \rho were rational.

Denote the kth digit of the binary expansion of \rho by r_k. Then since \rho is assumed rational, its binary expansion is eventually periodic, and so there exist positive integers N and k such that r_n = r_{n+ik} for all n N and all i \in \mathbb{N}.

Since there are an infinite number of primes, we may choose a prime p N. By definition we see that r_p=1. As noted, we have r_p=r_{p+ik} for all i \in \mathbb{N}. Now consider the case i=p. We have r_{p+i \cdot k}=r_{p+p \cdot k}=r_{p(k+1)}=0, since p(k+1) is composite because k+1 \geq 2. Since r_p \neq r_{p(k+1)} we see that \rho is irrational.

References

Weisstein, Eric W.. "Prime Constant".
Hardy, G. H.. (2008). "An introduction to the theory of numbers". Oxford University Press.

Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

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