#diophantine-equations

Find a rectangular box whose three edges, three face diagonals and space diagonal all have integer lengths, or prove that none exists. In other words, find positive integers $a,b,c$ such that $a^2+b^2$, $b^2+c^2$, $a^2+c^2$ and $a^2+b^2+c^2$ are all perfect squares.

Prove that for every integer $n\ge 2$ there are positive integers $x,y,z$ with $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$

Prove that the equation $a^n+b^n=c^n$ has no solution in positive integers $a,b,c$ when $n$ is an integer greater than 2.

The numbers $8=2^3$ and $9=3^2$ are consecutive perfect powers. Prove that they are the only such pair, that is, the only solution of $x^a-y^b=1$ in integers $x,y,a,b>1$ is $3^2-2^3=1$.