Consider the non-negative natural numbers:


Encode the primes as $1$, the rest as $0$.

$E = 0,0,1,1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1\dots$

The number $E_0$ is

$001.10101000101000101\dots$ (i.e. the binary pattern of the primes)

$E_0$ is irrational in every base due to a simple proof which shows that the binary pattern of the primes, $E$, is not periodic.

Since the location of the (binary) point is arbitrary, if we locate the point one place to the right, we obtain $E_1 (=0011.0101\dots)$.

So $\dfrac{E_1}{E_0}$ is the integer $2$.

Since $E_0$ is irrational in every number base, we are led to the interesting aphorism:

"Every integer greater than one is the ratio of two irrational numbers. Each of these irrational numbers expresses the infinite pattern of the prime numbers".

Now, prove that $E_0$ is transcendental.

Note that $C_0 = 110.01010111010111010\dots$ (i.e. the binary pattern of the composites)

and $E_0+C_0 = 111.11111111111111111\dots$

All contributions acknowledged.



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