Last week I was thinking about numbers while stuck in traffic, and I thought of this number:
x = sum 2^-p_k over k = 1, 2, 3, … and where p_k is the k-th prime number,
which is approximately 0.0110101000101000101000100000101… in base two and 0.41468250985111… in base ten.
Based on the fact that it has a non-repeating binary expansion, I know it’s irrational. But is it also transcendental?
I googled around and found Liouville numbers. I think this number is a Liouville number and would therefore be transcendental? Am I correct?
Given the fact that I thought of this number in a spare moment, surely some real mathematician has already documented it. Is there a proper name for my number? (Otherwise, I call dibs!)