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Problem: How many positive integers are solutions of the equation:

$\frac{1}{x}$ + $\frac{2}{y}$ = $\frac{3}{19}$

Comments:

  • Since the problem asks for positive integer solutions, I think I'm correct in saying that this is a Diophantine equation
  • The solution given to the problem, involves getting rid of the fractions, manipulating the expression so that it is factorable, and then looking what possible factors could work. (I'm not sure if that makes sense - if it doesn't I can attach the full given solution here).
  • My question is this: Is there are general method you would follow or any tips/tricks/hints you could offer to someone who is new to solving Diophantine equations? How do you know what to do? Is there a step-wise process that could be applied to other questions? If there are steps to follow, then maybe someone could also post another Diophantine equation and I can see if I can apply the 'step wise process' to that problem.

Many thanks!

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    $\begingroup$ $\dfrac{1}{x}$ + $\dfrac{2}{y}$ = $\dfrac{3}{19}\implies (3 x - 19) (3 y - 38) = 722$, this equation have 3 positive integers solution. $\endgroup$
    – Dmitry Ezhov
    Oct 17, 2020 at 7:51

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There are many, many kinds of diophantine equations. And in a precise sense there is no general method that works for all of them. Three tips or tricks that often help:

  1. Clear all denominators, leaving only sums and products of integers.
  2. Try to express the equation as a product. Then you can use unique factorization.
  3. Try to find a contradiction by considering the equation modulo $n$ for some number $n$.

The first two tricks are helpful here: As suggested in the comments, you first show that $$19y+38x=3xy,$$ which can be expressed as a product of the form $$(ax+b)(cy+d)=e,$$ for some integers $a$, $b$, $c$, $d$ and $e$. Comparing the two equations shows that it must be $$(3x-19)(3y-38)=722.$$ Now because $722=2\times19^2$ this leaves very few options for $3x-19$ and $3y-38$.

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  • $\begingroup$ Thanks @Servaes - Just two questions: Firstly, how did you go from the general form (ax+b)(cy+d) = e to the factored form (3x-19)(3x-38) =722? My second question is this: would you have an example of a Diophantine equation when the third trick is useful? $\endgroup$
    – Rushi
    Oct 17, 2020 at 10:12
  • $\begingroup$ Expanding the product shows that $ac=3k$, $ad=38k$, $bc=19k$ for some integer $k$. It is not hard to see that $k$ should be a multiple of $3$, and for $k=3$ you quickly find these coefficients. $\endgroup$
    – Servaes
    Oct 17, 2020 at 10:17
  • $\begingroup$ As for your second question; there are many examples of such questions on this site. The tags elementary-number-theory and number-theory, perhaps in combination with modular-arithmetic, should help you find some. A simple example would be the diophantine equation $$x^3-17y^3=5.$$ $\endgroup$
    – Servaes
    Oct 17, 2020 at 10:18

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