Prove that there exists infinitely many positive integer solutions in $(x,y)$ to the equation : $$\frac{x+1}{y} + \frac{y+1}{x} = 4$$
2026-03-25 20:36:02.1774470962
Infinitely many solutions of the equation $\frac{x+1}{y}+\frac{y+1}{x} = 4$
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$$\frac{x+1}{y} + \frac{y+1}{x} = 4$$
$$x^2+x+y^2+y=4xy$$
$$(x+y)^2+(x+y)=6xy$$
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?