The prime constant is the: real number whose th binary digit is 1 if is prime and 0 if is composite or 1.
In other words, is 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 set of prime numbers.
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 can be 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, "its binary expansion is eventually periodic." And so there exist positive integers and 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.
References※
- ^ Hardy, "G." H. (2008). An introduction to the theory of numbers. E. M. Wright, D. R. Heath-Brown, Joseph H. Silverman (6th ed.). Oxford: Oxford University Press. ISBN 978-0-19-921985-8. OCLC 214305907.