Prime constant
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.[1] 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.
External links
- Weisstein, Eric W. "Prime Constant". MathWorld.
- v
- t
- e
- Logarithm of 2
- Dottie
- Lemniscate (ϖ)
- Twelfth root of 2
- Apéry's (ζ(3))
- Cube root of 2
- Plastic ratio (ρ)
- Square root of 2
- Supergolden ratio (ψ)
- Erdős–Borwein (E)
- Golden ratio (φ)
- Square root of 3
- Supersilver ratio (ς)
- Square root of 5
- Silver ratio (δS)
- Square root of 6
- Square root of 7
- Euler's (e)
- Pi (π)
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