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?


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Question Number 216538 by CrispyXYZ last updated on 10/Feb/25

$Find all integer solutions of$ $3^{m} = 2 n^{2} + 1 .$ $I o n l y f o u n d m = 1, 2, 5 b y c o m p u t e r$ $f r o m m = 1 t o m = 30000 .$ $I s t h e r e a n y g r e a t e r s o l u t i o n s ?$
Commented by Rasheed.Sindhi last updated on 10/Feb/25

$A l s o m = 0$
Commented by Rasheed.Sindhi last updated on 12/Feb/25

$D e e p S e e k a s s u r e s t h a t t h e r e i s$ $n o o t h e r s o l u t i o n f o r m > 5$ $F i n a l A n s w e r :$ $(0, 0), (1, \pm 1), (2, \pm 2), (5, \pm 11)$
Commented by CrispyXYZ last updated on 12/Feb/25

$O k a y t h a n k s . A c t u a l l y I w a s s e e k i n g$ $p r o o f m e t h o d s .$
Commented by ArshadS last updated on 12/Feb/25

$d e e p s e e k a l s o c a n h e l p i n t h i s c o n n e c t i o n .$
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