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.
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.