Question Number 56800 by mr W last updated on 24/Mar/19
$${x},{y},{z}\:{are}\:{positive}\:{integers}. \\ $$$${find}\:{all}\:{solutions}\:{of}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{1}={xyz}. \\ $$
Commented by MJS last updated on 24/Mar/19
$$\mathrm{with}\:{z}=\mathrm{3}\:\mathrm{we}\:\mathrm{get}\:\mathrm{pairs}\:\mathrm{of}\:\mathrm{Fibonacci}\:\mathrm{numbers} \\ $$$$\mathrm{for}\:\left({x},\:{y}\right)\:\mathrm{or}\:\left({y},\:{x}\right)\:\mathrm{with}\:\mathrm{each}\:\mathrm{2}^{\mathrm{nd}} \:\mathrm{one}\:\mathrm{missing} \\ $$$$\left(\mathrm{1},\:\mathrm{2}\right) \\ $$$$\left(\mathrm{2},\:\mathrm{5}\right) \\ $$$$\left(\mathrm{5},\:\mathrm{13}\right) \\ $$$$\left(\mathrm{13},\:\mathrm{34}\right) \\ $$$$\left(\mathrm{34},\:\mathrm{89}\right) \\ $$$$... \\ $$
Commented by mr W last updated on 25/Mar/19
$${thank}\:{you}\:{sir}!\:{i}\:{need}\:{time}\:{to}\:{come} \\ $$$${behind}\:{it}.\:{is}\:{z}=\mathrm{3}\:{the}\:{only}\:{solution}? \\ $$
Commented by MJS last updated on 25/Mar/19
$$\mathrm{I}\:\mathrm{found}\:\mathrm{no}\:\mathrm{other}\:\mathrm{solution}\:\mathrm{but}\:\mathrm{I}'\mathrm{m}\:\mathrm{not}\:\mathrm{sure}... \\ $$