<s>[INST] number of solutions to an equation? Given $x$ and $y$ are multiples of $2$ satisfying 
$$x^2 - y^2 = 27234702932$$ 
Find the number of solutions to $x$ and $y$.
 [/INST] As $27234702932=2^2\cdot181\cdot37616993$ where the last two factors are primes
If $x=2X,y=2Y$
$$X^2-Y^2=181\cdot37616993$$
If $X^2-Y^2=p\cdot q$ where $p,q$ are primes, 
the possible cases for $X+Y,X-Y$ are $\pm pq, \pm 1$  and $\pm p,\pm q $
For example,  if $X+Y=1,X-Y=pq$
The number of positive factors of $p\cdot q$ is $(1+1)\cdot(1+1)=4$
So, there should be $4\cdot2 =8$ solutions in integers.
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