Computable, but is it transcendental?

[Tom K]

A real number, r, is the sum of a infinite sequence of rationals s.t. the
nth rational is either 0 (if n is composite)

or 10^-n (if n is
prime).

Since each step is computable r is computable, 0 < r < 1.

But is r transcendental? (I strongly suspect it is.)

Is there a test to determine if a digit has an equiprobable chance of being
in the nth position of a digital expansion of a real?

____

P.S. My interest here relates to the possibility that there is a subset of
transcendental (& possibly non-computational) reals that have digital
expansions which exhibit random-like qualities vis a vis equiprobability of
the aforementioned digits.

[Barry Schwarz]

Post by Tom K
A real number, r, is the sum of a infinite sequence of rationals s.t. the
nth rational is either 0 (if n is composite)
or 10^-n (if n is
prime).

What is the decision reached by the above rules for n = 0 and n = 1?

Post by Tom K
Since each step is computable r is computable, 0 < r < 1.
But is r transcendental? (I strongly suspect it is.)

Shouldn't the first question be is it rational or irrational? Has
someone proven that the intervals between successive primes does not
form a repeating pattern?

Post by Tom K
Is there a test to determine if a digit has an equiprobable chance of being
in the nth position of a digital expansion of a real?
____
P.S. My interest here relates to the possibility that there is a subset of
transcendental (& possibly non-computational) reals that have digital
expansions which exhibit random-like qualities vis a vis equiprobability of
the aforementioned digits.

[Tom K]

Post by Barry Schwarz
Shouldn't the first question be is it rational or irrational? Has
someone proven that the intervals between successive primes does not
form a repeating pattern?

The proof that r is irrational is that one can prove that for any integer,
say g, there exist g consecutive, composite numbers. Therefore, by reductio
ad absurdum, r cannot have a repeating expansion & hence is irrational. The
proof regarding g is covered in elementary, number theory courses so I did
not post it here.

To prove that r is transcendental is less straightforward, I believe.

[Tom K]

Post by Barry Schwarz

Post by Tom K
A real number, r, is the sum of a infinite sequence of rationals s.t. the
nth rational is either 0 (if n is composite)
or 10^-n (if n is
prime).

What is the decision reached by the above rules for n = 0 and n = 1?

Since there is no zeroth rational, n > 0.

As for the definition of the sequence I was initially going to use
_non-prime_ instead of composite, so for n = 1, the first rational was going
to be 0, but it could just as well be 1 (it really makes no important
difference). Thanks for catching my mistake.

[Tom K]

Post by Tom K
but it could just as well be 1 (it really makes no important
difference). Thanks for catching my mistake.

Erratum: please read 10^-1 for 1 above. Thank you.