Question Number 216538 by CrispyXYZ last updated on 10/Feb/25
$$\mathrm{Find}\:\mathrm{all}\:\mathrm{integer}\:\mathrm{solutions}\:\mathrm{of} \\ $$$$\mathrm{3}^{{m}} =\mathrm{2}{n}^{\mathrm{2}} +\mathrm{1}. \\ $$$$ \\ $$$${I}\:{only}\:{found}\:{m}=\mathrm{1},\:\mathrm{2},\:\mathrm{5}\:{by}\:{computer} \\ $$$${from}\:{m}=\mathrm{1}\:{to}\:{m}=\mathrm{30000}. \\ $$$${Is}\:{there}\:{any}\:{greater}\:{solutions}? \\ $$
Commented by Rasheed.Sindhi last updated on 10/Feb/25
$${Also}\:{m}=\mathrm{0}\: \\ $$
Commented by Rasheed.Sindhi last updated on 12/Feb/25
$${DeepSeek}\:{assures}\:{that}\:{there}\:{is} \\ $$$${no}\:{other}\:{solution}\:{for}\:{m}>\mathrm{5} \\ $$$${Final}\:{Answer}: \\ $$$$\:\:\left(\mathrm{0},\mathrm{0}\right),\left(\mathrm{1},\pm\mathrm{1}\right),\left(\mathrm{2},\pm\mathrm{2}\right),\left(\mathrm{5},\pm\mathrm{11}\right) \\ $$
Commented by CrispyXYZ last updated on 12/Feb/25
$${Okay}\:{thanks}.\:{Actually}\:{I}\:{was}\:{seeking} \\ $$$${proof}\:{methods}. \\ $$
Commented by ArshadS last updated on 12/Feb/25
$${deepseek}\:{also}\:{can}\:{help}\:{in}\:{this}\:{connection}. \\ $$