Solving Diophantine Equations? What are the common techniques?

Question

What does it mean to solve a Diophantine Equation? Some questions are like

Solve $x^2+1=2y^4$ over the integers.

What does it actually mean to solve a diophantine equation? Do we have to find a single solution or an infinite number of them? Can solving mean that we have to prove that none of them exist?

Some Diophantine Equations are very complicated.

Also, if we have to find solutions, what are the techniques involved in solving Diophantine Equations? By techniques, I mean some procedures or ideas to solve them. Which are the common techniques employed while solving them? I need a list of techniques that I can try while solving a Diophantine Equation and I believe that I have a very little knowledge about these techniques.

Answer 1

1. To solve a diophantine equation means to find such (x,y) so that the base equation problem holds (like in your example given). If no special domain is strictly given, then normally one understands that (x,y) should be integers (discrete solution). However, rational solutions are sometimes also considered.

2. As regards the number of solutions -- it depends: some diophantine equations may have no solutions over integers or rationals, some may have one unique solution, some may have entire families of solutions. Note, that "diophantine equations" is a name for a gigantic class of equations with highly diverse complexity and potential applications/interpretations.

3. As regards solving techniques, again - this depends on what particular subclass of diophantine equation you think about. For a start I think it's good to see how general quadratic diophantine equations can be solved. See here: https://www.alpertron.com.ar/METHODS.HTM for reference. A very good article is also: http://www.numbertheory.org/pdfs/general_quadratic_solution.pdf