# 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::

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.

• Evgeny.Makarov

#### 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...