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My motivation for the question is the following question: Why is it so difficult to determine whether or not $\Large \pi^{\pi^{\pi^\pi}}$ is an integer?

I mean before asking Why is it so difficult to determine whether or not $\Large \pi^{\pi^{\pi^\pi}}$ is an integer?, what about the simpler one which is $\Large \pi^{\pi^{\pi}}$.

If we were to try compute this number and prove that it lies strictly between two integers meaning it's not integer, to how many digits should $\pi$ be approximated?

Math Admiral
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    I suspect this number isn’t so big, so we actually can handle the calculations and show it is a non-integer. WolframAlpha says it’s about $1.34 \times 10^{18}$. – Robin Apr 29 '25 at 20:01
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    …indeed it gives it in the form $a \times 10^{18}$ where $a$ has more than $18$ given decimal places – Robin Apr 29 '25 at 20:02
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    $1340164183006357435.297449129640131415\ldots$ – Henry Apr 29 '25 at 20:04
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    $\Large{ \pi^{\pi^{\pi}} }=1340164183006357435.297449\ldots$ is clearly not an integer, but $\Large \pi^{\pi^{\pi^\pi}} = 10^{666262452970828652.4172\ldots}$ has $666262452970828653$ digits before the decimal point and so is much harder to calculate – Henry Apr 29 '25 at 20:10

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According to Wolfram Alpha, its value is $1.34016418300635743529744912964013141509937497457349923778792751658603... × 10^{18}$ with continued fraction $[1340164183006357435; 3, 2, 1, 3, 4, 1, 1, 5, 1, 1, 1, 4, 14, 1, 2, 5, 2, 3, ...]$ so it pretty clearly is not an integer.

marty cohen
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    The number may be easier to understand if you shift the decimal point 18 places to get $1340164183006357435.297449\dots$. – Dan Apr 29 '25 at 20:08