solve equation in positive integers Can anybody help me with this equation?
Solve in $\mathbb{N}$:
$$3x^2 - 7y^2 +1=0$$
One solution is the pair $(3,2)$, and i think this is the only pair of positive integers that can be a solution. Any idea?

For some Pell equations. For special cases, there is a formula describing their solutions.
If the equation: $aX^2-qY^2=f$ rational root $\sqrt{\frac{f}{a-q}}$
Solutions can be written using the following equation Pell:  $p^2-aqs^2=1$
Solutions have the form:
$Y=(2aps\pm(p^2+aqs^2))\sqrt{\frac{f}{a-q}}$
$X=(2qps\pm(p^2+aqs^2))\sqrt{\frac{f}{a-q}}$
According to these decisions can be found double this solution:
$Y_2=Y+2as(Yqs-Xp)$
$X_2=X+2p(Yqs-Xp)$
If the other root is rational: $\sqrt{\frac{f}{a}}$
Solution has the form:
$Y=2ps\sqrt{fa}$
$X=(p^2+aqs^2)\sqrt{\frac{f}{a}}$
Must take into account that the number: $p,s$ can be any character.
