Question Number 195413 by Calculusboy last updated on 02/Aug/23
Commented by Rasheed.Sindhi last updated on 02/Aug/23
$${x}^{{y}} +{y}^{{x}} =\mathrm{8}\Rightarrow{x}={y}=\mathrm{2} \\ $$$${x}+{y}={k}^{\mathrm{2}} \Rightarrow{k}^{\mathrm{2}} =\mathrm{2}+\mathrm{2}=\mathrm{4}\Rightarrow{k}=\pm\mathrm{2} \\ $$$${k}+\mathrm{1}=\pm\mathrm{2}+\mathrm{1}=\mathrm{3},−\mathrm{1} \\ $$
Commented by TheHoneyCat last updated on 03/Aug/23
quite a lazy and un-rigorous proof by Rasheed Sindhi, (for instance x=y=2 is not the only solution, you also have (1,7)) but (assuming we are only looking for integer values of k), his final results is correct. And easy to verify. Hence I am promoting it to an answer, if anyone wants details, please ask.
Commented by Rasheed.Sindhi last updated on 03/Aug/23
$${Yes}\:{sir}\:{my}\:{answer}\:{is}\:{not}\:{a}\:{complete} \\ $$$${solution}. \\ $$
Commented by TheHoneyCat last updated on 03/Aug/23
Yes, no problem. I was just justifying "promoting it to answer" because I think its sufficient for anyone to find the details on their own, starting from what you wrote.
Commented by Rasheed.Sindhi last updated on 04/Aug/23
$${See}\:{answer}\:{of}\:{mr}\:{W}\:{sir}\:{to}\:{the} \\ $$$${Q}#\mathrm{195511} \\ $$