perfect square

jacks

Well-known member
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::

$\hspace{-16}Let \bf{x^2+3y=k^2} and \bf{y^2+3x=l^2}\\\\ Where \bf{x,y,k,l\in \mathbb{Z^{+}}}\\\\ \bf{(x^2-y^2)-3(x-y)=k^2-l^2}\\\\ \bf{(x-y).(x+y-3)=(k+l).(k-l)}\\\\ \bullet\;\; \bf{(x-y)=k+l\;\;,(x+y-3)=k-l}\\\\ \bullet\;\; \bf{(x-y)=k-l\;\;,(x+y-3)=k+l}\\\\ So \bf{x=\frac{2k+3}{2}\notin \mathbb{Z^{+}}}\\\\ and \bf{y=\frac{-2l+3}{2}\notin \mathbb{Z^{+}}}\\\\$

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

Well-known member
MHB Math Scholar
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.

Jester

Well-known member
MHB Math Helper
what about $(1,1)$?

Jester

Well-known member
MHB Math Helper
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

In Memoriam
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...