I have a set of Diophantine equations which I know only one equation has a single solution.
I am trying to find a way to give probabilities to which equation contains the solution.
For example, which of the following most likely contains a solution?
16xy + 1x + 13y = 249835285
16xy + 3x + 15y = 249835283
16xy + 5x + 9y = 249835283
16xy + 7x + 11y = 249835281
Where 'x' and 'y' are both non-zero, positive integers.
I have tried using modular arithmetic to see how many possible solutions exist for each equation.
Eg
16xy + 1x + 13y = 249835285
1xy + 1x + 1y = 1 (mod 3)
1xy + 1x + 3y = 0 (mod 5)
2xy + 1x + 6y = 0 (mod 7)
But there appears to be equal possible solutions to each.
For the curious, x = 3148 and y = 4959 is the solution to 16xy + 5x + 9y = 249835283.
There's no probability involved: a particular equation either has positive integer solutions or it doesn't.
In general, $a x y + b x + c y = d$ (with $a,b,c,d$ positive integers) can be written as $$ (ax + c)(ay + b) = ad + bc $$ The solutions correspond to ways to factor $ad + bc = uv$ where $u \equiv c \mod a$ and $v \equiv b \mod a$, $u > c$ and $v > b$: then $x = (u-c)/a$ and $y = (v - b)/a$ are solutions.
In your first case, $a=16$, $b = 1$, $c = 13$, $d =249835285$, $ad+bc = 3997364573 = 50377 \times 79349$, where $50377$ and $79349$ are primes $\equiv 9$ and $5 \mod 16$ respectively. There is no acceptable choice for $u$ and $v$.
In your third case, $a=16$, $b=5$, $c=9$, $d=249835283$, $ad+bc = 3997364573 = 50377 \times 79349$ again, so this time we can take $u=50377$ and $v = 79349$.