Convert PI to base 4. Does my unique human genome exist in the sequence of digits?

Question

The human genome consists of sequences of BASE Pairs A G C T Convert the number PI to base 4. Does my unique human genome exist in the sequence of digits?

Answer 1

A heuristic argument would go as follows: Assume your genome $G$ is a string of $n\gg1$ digits over $\{0,1,2,3\}$. Denote by $x_k$ the $k^{\rm th}$ digit of $\pi$ in base $4$. For each $r\geq0$ the probability that $$(x_{rn+1},x_{rn+2},\ldots,x_{rn+n-1}, x_{(r+1)n})\ne G$$ amounts to $(4^n-1)/ 4^n<1$. Therefore the probability that for no $r\geq0$ we have a coincidence is $$\lim_{N\to\infty}\left({4^n-1\over 4^n}\right)^N=0\ .$$

Answer 2

The facts are these:

• nobody knows the answer for sure;
• every mathematician in the world expects that the answer is Yes.

But we don't have a proof one way or the other, in spite of much effort invested in the problem.