MHB Number of Positive Integer Pairs for Perfect Squares

[juantheron]

the number of ordered pairs of positive integers $x,$y such that $x^2 +3y$ and $y^2 +3x$

are both perfect squares

my solution::

http://latex.codecogs.com/gif.latex?\hspace{-16}$Let%20$\bf{x^2+3y=k^2}$%20and%20$\bf{y^2+3x=l^2}$\\%20Where%20$\bf{x,y,k,l\in%20\mathbb{Z^{+}}}$\\%20$\bf{(x^2-y^2)-3(x-y)=k^2-l^2}$\\%20$\bf{(x-y).(x+y-3)=(k+l).(k-l)}$\\%20$\bullet\;\;%20\bf{(x-y)=k+l\;\;,(x+y-3)=k-l}$\\%20$\bullet\;\;%20\bf{(x-y)=k-l\;\;,(x+y-3)=k+l}$\\%20So%20$\bf{x=\frac{2k+3}{2}\notin%20\mathbb{Z^{+}}}$\\%20and%20$\bf{y=\frac{-2l+3}{2}\notin%20\mathbb{Z^{+}}}$\\

no possibilities.

but there is also more possibilities

like $(x-y).(x+y-3) = 1 \times (k^2-l^2) = (k^2-l^2) \times 1$

My Question is that is any pairs for which $x^2+3y$ and $3x^2+y$ are perfect square

Thanks

 

[Evgeny.Makarov]

We have $x^2+3y=(x+a)^2$ for some positive integer $a$ and similar for $y$ and some $b$. Express $x$ and $y$ through $a$ and $b$ and see when $x$ and $y$ are positive integers.

 

[Mathwebster]

what about $(1,1)$?

 

[Mathwebster]

the number of ordered pairs of positive integers $x,$y such that $x^2 +3y$ and $y^2 +3x$

. . . .

My Question is that is any pairs for which $x^2+3y$ and $3x^2+y$ are perfect square

Thanks

I think you just changed the question.

 

[Wilmer]

My Question is that is any pairs for which $x^2+3y$ and $3x^2+y$ are perfect square

Of course; infinite:
1,1
2,4
3,9
4,16
5,25
...and on...