Find-all-integer-solutions-of-3-m-2n-2-1-I-only-found-m-1-2-5-by-computer-from-m-1-to-m-30000-Is-there-any-greater-solutions-

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

$${deepseek}\:{also}\:{can}\:{help}\:{in}\:{this}\:{connection}. \\ $$

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