TITLE: How can one prove that $\pi^4 + \pi^5 < e^6$?
QUESTION [15 upvotes]: A proof of the inequality using properties of $\pi$ and $e$, for example, is what I'm looking for. Not calculator approximations showing the inequality holds.

REPLY [5 votes]: You can do it only by approximations. In this case you can use that $\pi<3.14159267$ and $e>2.71828182$. So $\pi^4+\pi^5<3.14159267^4+3.14159267^5$ and $e^6>2.71828182^6$. Hence it suffices to prove that :$$\left(\frac{314159267}{100000000}\right)^4 + \left(\frac{314159267}{100000000}\right)^5 <\left(\frac{271828182}{100000000}\right)^6$$ After multiplication by $100000000^6$ and dividing by the common factor of 64 this is equivalent to:$$12500000^2*314159267^4 + 1562500*314159267^5 - 135914091^6<0$$
This is as good as it can get. Plug it in some calculation software to get the result...$-1766486976776637078635986938150126848541$. This may seem a large number but relatively speaking it is very small it is about $3*10^{-10}$ from $135914091^6$.