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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.

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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 $v = 79349$.

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  • $\begingroup$ What about if we don't know the factorization of ad+bc? Is there a method which makes one of the equations 'more likely' to lead to the solution? $\endgroup$ – threenplusone Jul 21 '16 at 7:25
  • $\begingroup$ As $n \to \infty$, the numbers of primes $\le n$ that are $\equiv b \mod a$ are approximately equal for all $b$ coprime to $a$, so in that sense none of the equations is very much "more likely" than the others. Of course those numbers won't be exactly equal. Thus of the $6335$ odd primes $\le \sqrt{3997364573}$, $1574$ are $\equiv 1$ or $13 \mod 16$ while $1579$ are $\equiv 5$ or $9 \mod 16$. So in that sense you might say that the third case is very slightly "more likely" than the first. $\endgroup$ – Robert Israel Jul 21 '16 at 17:11

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