Question Number 4362 by Rasheed Soomro last updated on 13/Jan/16
$$\mathrm{Determine}\:\mathrm{integers}\:\mathrm{x},\mathrm{y},\mathrm{z}\:\mathrm{satisfying}: \\ $$$$\mathrm{ax}^{\mathrm{b}} +\mathrm{by}^{\mathrm{c}} =\mathrm{cz}^{\mathrm{a}} \\ $$$$\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{are}\:\mathrm{fixed}\:\mathrm{integers}. \\ $$
Commented by Yozzii last updated on 14/Jan/16
$${x}={u}^{\mathrm{1}/{b}} ,{y}={k}^{\mathrm{1}/{c}} ,{z}={p}^{\mathrm{1}/{a}} \:\:\:\:{a},{b},{c},{x},{y},{z}\in\mathbb{Z} \\ $$$${au}+{bk}={cp}\: \\ $$