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(3n+1)^2+4n^3=m^2
Posted: Mon Feb 23, 2015 9:29 pm
by Phlembac Adib Hasan
Prove that there exist infinitely many positive integer $n$s such that $(3n+1)^2+4n^3$ is a perfect square.
Hint:
Re: (3n+1)^2+4n^3=m^2
Posted: Mon Feb 23, 2015 11:16 pm
by Nirjhor
For all $k\in\mathbb N$ we have $n=k^2+k\Rightarrow (3n+1)^2+4n^3=(2k+1)^2(k^2+k+1)^2$.
Re: (3n+1)^2+4n^3=m^2
Posted: Tue Mar 03, 2015 9:15 pm
by Mahfuz Sobhan
how should i think to solve this question? i.e the problem deals with which topic of math?
Re: (3n+1)^2+4n^3=m^2
Posted: Thu Mar 05, 2015 7:41 pm
by Phlembac Adib Hasan
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.