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 January 18th, 2019, 08:49 AM #1 Senior Member   Joined: Dec 2015 From: iPhone Posts: 387 Thanks: 61 Integer part How can I find the integer part of $\displaystyle e^e \;$? (without calculator) e - Euler constant What about approximation? First, I tried one example for $\displaystyle 2e$ and $\displaystyle e^2$, to see whether it is possible. $\displaystyle 3>e>\frac{5}{2}\;$ ,then multiply by 2 6>2e>5 so integer part of 2e is 5. Last edited by skipjack; January 19th, 2019 at 12:21 AM.
 January 19th, 2019, 12:27 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,298 Thanks: 1971 How much basic arithmetic is acceptable, given that use of a calculator isn't allowed? Can you assume that e = 2.718... or does that have to be proved as well?
 January 19th, 2019, 02:01 AM #3 Senior Member   Joined: Dec 2015 From: iPhone Posts: 387 Thanks: 61 By calculator ,integer part of $\displaystyle e^e$ is 15. But integer part of $\displaystyle {2.7}^{2.7}$ is 14 , same for 2.71 . Last edited by idontknow; January 19th, 2019 at 02:07 AM.
 January 19th, 2019, 05:56 AM #4 Global Moderator   Joined: Dec 2006 Posts: 20,298 Thanks: 1971 $e^{2.71} >15$, but I think a lot of arithmetic is needed to prove that.

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