Infinitely many solutions of the equation $\frac{x+1}{y}+\frac{y+1}{x} = 4$ Prove that there exists infinitely many positive integer solutions in $(x,y)$ to the equation :
$$\frac{x+1}{y} + \frac{y+1}{x} = 4$$

$$\frac{x+1}{y} + \frac{y+1}{x} = 4$$
$$x^2+x+y^2+y=4xy$$
$$(x+y)^2+(x+y)=6xy$$
Let $X=x+y$ and $Y=x-y$.  Then $X^2+X=\dfrac32\left(X^2-Y^2\right)$ or $(X-1)^2-3Y^2=1$.
That's a Pell equation, which has infinitely many solutions:
$X-1=$$1,2,7,26,97,...$ and $Y=$$0,1,4,15,56,...$.
Click on the blue numbers to see more.  Can you take it from here?
