prime constant

The prime constant is the real number $\rho$ whose $n$th 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,
:$\rho = \sum_ \frac = \sum_^\infty \frac$
where $p$ indicates a prime and $\chi_$ is the characteristic function of the set $\mathbb$ 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 $\rho$ can be shown to be irrational. To see why, suppose it were rational.

Denote the $k$th 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
$r_n = r_$ for all $n > N$ and all $i \in \mathbb$.

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 $r_p=r_$ for all $i \in \mathbb$. Now consider the case $i=p$. We have $r_=r_=r_=0$, since $p(k+1)$ is composite because $k+1 \geq 2$. Since $r_p \neq r_$ we see that $\rho$ is irrational.

References

External links

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{{DEFAULTSORT:Prime Constant
Irrational numbers
Prime numbers
Articles containing proofs
Mathematical constants