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please help !

Find all the solutions for k and L where both of them are positive integers and

3(2k+1)(2k-1)=L^2

Note by Gaurav Sharma
2 years, 1 month ago

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Hi;

It looks like a homework question and it appears that your book/teacher expects you to say that \[(k,L) = (1,3)\] is the only solution.

However, there are more solutions. The next solution is 141 digits.

1
2
k = 466642456252969429548984353760000301550329127624397271829679711032060980203808963148776334310821825712845861387066156068094839910543082337321
L = 1616496886397760388309674049191810513095994873514870571782730987813611003182466173777437593238208569111687366772629322638941777021682113779933 

To verify this, you'll need a Computer Algebra System or a multi-precision programming language.

Furthermore, I conjecture that there are infinite such solutions Agnishom Chattopadhyay · 2 years, 1 month ago

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@Agnishom Chattopadhyay @Gaurav Sharma , I am sorry. {k -> 181, L -> 627}, {k -> 13, L -> 45} also works. There are many small solutions too. Agnishom Chattopadhyay · 2 years, 1 month ago

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@Agnishom Chattopadhyay Thanks mahn ! Gaurav Sharma · 2 years, 1 month ago

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