Non-calculator proof that $\pi^\pi -\pi \lt \frac{100}{3}$ I am looking for a few non-computational, non-calculator proof of the following inequality:
$$\pi^\pi -\pi \lt \frac{100}{3}$$
I can't really seem to come up with a proof because of that killer $\pi^\pi$ term.
 A: The inequality $x^x-x<100/3$ is true precisely when $0<x<3.14175798627357...$, so intuitively a solution to this problem must in some fashion make use of the fact that $$\pi<3.14175798627357\ldots.$$ The upper bound $\pi<22/7$ is insufficient for this, but the upper bound $355/113$ suffices. (The bound of $355/113$ dates to the fifth century CE, which I personally would be prepared to admit as "non-calculator".) One could therefore proceed as follows: since $x \mapsto x^x-x$ is increasing for $x>1$ (an easy calculus exercise), it is sufficient to show 
$$\left(\frac{355}{113}\right)^{\frac{355}{113}}-\frac{355}{113}<\frac{100}{3}$$
which is to say
$$\left(\frac{355}{113}\right)^{\frac{355}{113}}<\frac{12365}{339}$$
or
$$3\times 355^{\frac{355}{113}}<12365\times 113^{\frac{242}{113}}$$
or
$$3^{113}\times 355^{355}<12365^{113}\times 113^{242}.$$
It is not immediately obvious to me that this inequality cannot be obtained by hand within a broadly viable timeframe, but before attempting it I would probably start reading up on things like Karatsuba multiplication, and work out carefully beforehand how many pages I might need to fill with long multiplication calculations and endless repeated squaring. The gap between the two numbers is very small, one being about $1.7\times 10^{959}$ and the other around $1.8\times 10^{959}$. In any event, I hope that the sharpness of the inequality which needs to be proved illustrates the difficulty of the problem.
