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

#Algebra #HelpMe! #Help #Pleasehelp

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Comments

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 - 6 years, 6 months ago

@Gaurav Sharma , I am sorry. {k -> 181, L -> 627}, {k -> 13, L -> 45} also works. There are many small solutions too.

Agnishom Chattopadhyay - 6 years, 6 months ago

Thanks mahn !

Gaurav Sharma - 6 years, 6 months ago

@Calvin Lin @Krishna Ar @Agnishom Chattopadhyay

Gaurav Sharma - 6 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$