How many pairs of integers are there such that $$\huge a^2+3b^2$$ and $$\huge b^2+3a^2$$ are perfect squares?

Question

How many pairs of integers are there such that $$\huge a^2+3b^2$$ and $$\huge b^2+3a^2$$  are perfect squares?

anonymous · Answer

i proved both of them to even or odd

anonymous · Answer

@satellite73 ?? @Callisto ??? @shadowfiend ?? @ParthKohli ??@Directrix ??

parthkohli · Answer

Brilliant.org?

parthkohli · Answer

I've seen such a problem, except the 3 was a 6.

parthkohli · Answer

$$a=b$$@shubhamsrg ;-)

shubhamsrg · Answer

shubhamsrg · Answer

shouldn't there be infinite solutions for this?
when a=b
we have 4a^2

for various values of a(or b), this is always a perfect square! :O

anonymous · Answer

no there are not infinite solutions!

shubhamsrg · Answer

where am I going wrong in my reasoning ?

anonymous · Answer

1,1 can be a solution by hit and trial!