(3n+1)^2+4n^3=m^2

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Phlembac Adib Hasan
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(3n+1)^2+4n^3=m^2

Unread post by Phlembac Adib Hasan » Mon Feb 23, 2015 9:29 pm

Prove that there exist infinitely many positive integer $n$s such that $(3n+1)^2+4n^3$ is a perfect square.
Hint:
Try some small values
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Nirjhor
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Re: (3n+1)^2+4n^3=m^2

Unread post by Nirjhor » Mon Feb 23, 2015 11:16 pm

For all $k\in\mathbb N$ we have $n=k^2+k\Rightarrow (3n+1)^2+4n^3=(2k+1)^2(k^2+k+1)^2$.
- What is the value of the contour integral around Western Europe?

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Mahfuz Sobhan
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Re: (3n+1)^2+4n^3=m^2

Unread post by Mahfuz Sobhan » Tue Mar 03, 2015 9:15 pm

how should i think to solve this question? i.e the problem deals with which topic of math?

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Re: (3n+1)^2+4n^3=m^2

Unread post by Phlembac Adib Hasan » Thu Mar 05, 2015 7:41 pm

Typically these equations are called Diophantine Equations. And it's part of Number Theory. I don't know if there is any good (olympiad level) material on this topic. Try "Introduction to Diophantine Equations", perhaps?

Also, in this thread, you are not required to solve that equation. It was actually an interesting factorization (that saved my life) while working on some other sh*ts.
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