Is it true $\sqrt{2^{3^{-5^{-7^{11^{13^{-17^{-19^{23^{29^{-31^{-37^{41^{\ldots}}}}}}}}}}}}}} =\sqrt{2}$?

Question

Question: Is it true and easy to show that:

$$\sqrt{2^{3^{-5^{-7^{11^{13^{-17^{-19^{23^{29^{-31^{-37^{41^{\ldots}}}}}}}}}}}}}} =\sqrt{2}$$ ?

Implicitly I am making the claim $2^{3^{-5^{-7^{11^{13^{-17^{-19^{23^{29^{-31^{-37^{41^{\ldots}}}}}}}}}}}}} =2$. Computationally it has been verified up to the prime exponent 41. I do not have a computing platform to handle additional exponents. Note it is not an alternating sequence of $+/-$. But rather the parity in the prime exponents reads starting at the exponent $3$: $+,-,-,+,+,-,-,+,+,-,-,\ldots$. I have no reason to believe it is true on the other hand I do not believe it coincidence.

Background and motivation: I was looking for "easy", possibly trivial, expression in order to rewrite the $\sqrt{2}.$ For example it is apparently easy to show that $\sqrt{\sqrt{3-\sqrt{3+\sqrt{3-\sqrt{3+\ldots}}}}}=\sqrt{2}.$ So, I began "playing" with other expression in Wolfram Alpha and came up with this question. In order to convince yourself just consider typing $\text{sqrt(2^3^-5^-7^11^13^-17^-19^23^-29^-31^37)}$ into the google search bar. I was worried this was obvious but I cannot seem to figure it out. I don't think it is computational junk from Wolfram or Google.

Answer 1

Convergence of your tower to $2$ is equivalent to convergence of $3^{-5^{-7^{11^\ldots}}}$ to $1$, which is equivalent to convergence of $5^{-7^{11^{13^\ldots}}}$ to $0$, which is equivalent to convergence of $7^{11^{13^{-17^\ldots}}}$ to $\infty$, which is equivalent to convergence of $11^{13^{-17^{-19^\ldots}}}$ to $\infty$, which is equivalent to convergence of $13^{-17^{-19^{23^\ldots}}}$ to $\infty$, which is equivalent to convergence of $17^{-19^{23^{29^\ldots}}}$ to $-\infty$, which is absurd.

Answer 2

I'll add to Hagen von Eitzen's answer. If we truncate the power tower at any point past $29$, then we have the following inequalities:

\begin{align*} 1 &\le 23^{29^\cdots} \\ 19 &\le 19^{23^\cdots} \\ 0 &\le 17^{-19^\cdots}\le 17^{-19} \\ 13^{-17^{-19}} &\le 13^{-17^\cdots} \le 1 \\ 11^{13^{-17^{-19}}} &\le 11^{13^\cdots} \le 11 \\ 7^{11^{13^{-17^{-19}}}} &\le 7^{11^\cdots} \le 7^{11} \\ 5^{-7^{11}} &\le 5^{-7^\cdots} \le 5^{-7^{11^{13^{-17^{-19}}}}} \\ 3^{-5^{-7^{11^{13^{-17^{-19}}}}}} &\le 3^{-5^\cdots} \le 3^{-5^{-7^{11}}} \\ 2^{3^{-5^{-7^{11^{13^{-17^{-19}}}}}}} &\le 2^{3^\cdots} \le 2^{3^{-5^{-7^{11}}}} \end{align*}

You can check that $2^{3^{-5^{-7^{11}}}} < 2$, so it is impossible for the power tower to converge to $2$, but the partial towers are extremely close to $2$. You can check that both $\log_{10}\left(2-2^{3^{-5^{-7^{11^{13^{-17^{-19}}}}}}}\right)$ and $\log_{10}\left(2-2^{3^{-5^{-7^{11}}}}\right)$ are very roughly $-1.382 \times 10^9$.