Question

Prove that the equation $\frac{x+1}{y}+\frac{y+1}{x}=4$has infinitely many solutions in positive integers.

Answer 1

Suppose that the equation has a solution $(x_1,y_1)$ with $x_1\le y_1$
Clearing denominators, we can write the equation under the form $x^2-(4y-1)x+y^2+y=0$
that is, a quadratic in $x$. One of the roots is $x_1$, therefore. by Vieta's theorem, the second one is $4y_1-1-x_1$.
Observe that $4y_1-1-x_1\ge 4x_1-1-x_1=3x_1-1\ge 2x_1>0$
Hence, $4y_1-1-x_1$ is a positive integer. It follows that $(4y_1-1-x_1,y_1)$
is another solution of the system, Because the equation is symmetric, we obtain that $(x_2+y_2)=(y_1,4y_1-1-x_1)$
is also a solution.To end the proof, observe that $x_2+y_2=5y_1-1-x_1+y_1$
and that (1,1) is a solution. Thus , we can generate infinitely many solutions: $(1,1)\to (1,2)\to (2,6)\to$