Which precision would be needed?

Question

According to Wikipedia, it is not known whether the number

$$\pi \uparrow \uparrow 4$$

is an integer.

(See Tetration)

To which precision would $\pi$ have to be calculated to decide this ?

The given number has magnitude roughly

$$10 \uparrow 10 \uparrow 18$$

Answer 1

Writing a simple FORTRAN code, with quadruple precision, I got that

$\log_{10}(\pi\uparrow\pi\uparrow\pi\uparrow\pi)=666262452970848503.958096488385085\ldots$.

This means that $\lfloor\pi\uparrow\pi\uparrow\pi\uparrow\pi\rfloor$ is a positive integer with $N=666262452970848504$ digits. So you need to approximate $\pi\uparrow\pi\uparrow\pi\uparrow\pi$ with relative error less that $10^{-N}$ - Most of computing is done with $10^{-16}$ to $10^{-32}$ relative error.

Answer 2

OK, but my question was, which precision is needed for $\pi$.

Actually, the results come out very nearly the same, but just for the sake of argument, one can get more exact results by calculating $(\pi+x)\uparrow\uparrow 4$, in terms of $\pi\uparrow\uparrow 4$. I used the following general equations; $\exp(a+x)=\exp(a)(1+x+Ox^2)$, and $\ln(a+x)=\ln(a(1+\frac{x}{a}))=\ln(a)+\frac{x}{a}+Ox^2$. Then with a little algebra, by taking the logarithm, and then the exponenent, one gets the following very useful general result:

$$(A+ax)^{B+bx} = A^B(1+x(b\ln(A) + \frac{aB}{A})+Ox^2)$$

Now substitute $A=\pi+x$, $B=\pi+x$.
$$(\pi+x) ^ {\pi + x} = (\pi ^ \pi )(1+x(\ln(\pi) + 1)+Ox^2)$$

Now substitute $A=\pi+x$, $B=\pi^\pi, b=x\pi^\pi(\ln(\pi)+1)$. $$(\pi+x)\uparrow \uparrow 3 = (\pi \uparrow \uparrow 3)(1+x(\pi\uparrow\uparrow2)((1+\ln(\pi)) \ln(\pi)+\frac{1}{\pi}) +Ox^2)$$

For the next step, the equations are pretty messy, but numerical results are straightforward and the form of the result is as follows with $k\approx 3.18357671687997992229145094$

$$(\pi+x)\uparrow \uparrow 4 = (\pi \uparrow \uparrow 4)(1+x(\pi \uparrow\uparrow 3)(\pi\uparrow\uparrow2)k + Ox^2)$$

Now, for the number of digits we get the same as the other answer, $\log_{10}(\pi\uparrow\uparrow 4) = \log_{10}(\pi) (\pi \uparrow\uparrow 3) \approx 666262452970848503.958$

And finally, for the required precision of $\pi$, one takes the $\log_{10}$ of the coefficient of x, using the same numerical value of k as above, and then one gets $$\log_{10}((\pi\uparrow\uparrow 4) (\pi\uparrow\uparrow 3) (\pi\uparrow\uparrow 2) k ) = \log_{10}(\pi) (\pi \uparrow\uparrow 3 + \pi \uparrow\uparrow 2 + \pi) + \log_{10}(k) \approx 666262452970848524.150$$

This is about 20.2 digits additional precision required for $\pi$, than the number of digits in $\pi \uparrow \uparrow 4$. One would need a few more digits of precision than that, to calculate several additional decimal digits, to verify that $\pi \uparrow \uparrow 4$ is not an exact integer.