<s>[INST] Find all possible positive integers $x$ and $y$ such that the equation: $(x+y)(x-y)=\frac{(y+1)(y-1)}{24}$ is satisfied. My approach so far:
The given equation can be rewritten as: $x^2 -y^2=\frac{y^2 -1}{24}.$ This gives
$24x^2 +1=25y^2=(5y)^2.$ So $(24x^2+1)$ must also be a perfect square. This implies $x=0, 1$ is two such possible values for $x$. As $x>0$, so $x=1$ is a possible solution. Corresponding to this, we get $y=1$. But how to check does there exist any other solutions or not? Please suggest.. Thanks in advance. 
 [/INST] $x^2-Dy^2=1$
Let $(x_0,y_0)$ is  the smallest nontrivial positive solution.
Then general solution is given as follows.
$x_n + y_n\sqrt{D} = (x_0 + y_0\sqrt{D})^n$ 
Let $X=5y$ and $Y=x$ then $24x^2 - 25y^2 =- 1$ is transformed to $X^2-24Y^2=1.$
Smallest nontrivial positive solution is $(X0,Y0)=(5,1).$
Then all solutions are given $X_n + Y_n\sqrt{D} = (5 + 2\sqrt{6})^n.$
We show ten solutions with $n=1\cdots 10.$
$[5+2\sqrt{6}], [49+20\sqrt{6}], [485+198\sqrt{6}], [4801+1960\sqrt{6}], [47525+19402\sqrt{6}], [470449+192060\sqrt{6}], [4656965+1901198\sqrt{6}], [46099201+18819920\sqrt{6}], [456335045+186298002\sqrt{6}], [4517251249+1844160100\sqrt{6}]$
Since $X_n$ must be divisible by $5$, we know $[485+198sqrt{6}]$ is second solution.
Thus $y=\frac{485}{5}=97, x=\frac{198}{2}=99.$
Third solution is $y=\frac{47525}{5}=9505, x=\frac{19402}{2}=9701.$ 
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