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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 From Wikipedia, the free encyclopedia

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The prime constant is the real number whose th binary digit is 1 if is prime and 0 if is composite or 1.[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)[1]

The beginning of the binary expansion is: (sequence A010051 in the OEIS)

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Irrationality

The number is irrational.[2]

Proof by contradiction

Suppose 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.

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References

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