Diophantine equations of fourth degree

Question

Lets have the equation $X^4=DY^4+A^4$. Does anyone know how to obtain all the solutions of this equation? All numbers non zero naturals.

Answer 1

Let me tell what I know about such things.

Before telling you that let me suggest the ways to ask questions and receive proper answers here, if you want a proper response you need to take care of formatting, and also the punctuation marks, mistakes are common for all, everyone must learn these things one or the other time. Trust me ! , I have still faced more worse situations but changed myself a bit in formatting. And also be prompt in choosing the answer. If you don't choose the answer it annoys the person who answered it by taking strains and wasting their valuable time ( those things don't refer to me but to many great people present here ok ? )

To start your question is very vague , I can see that $X^4=DY^4+A^4$ is a degree four equation, so there are two possibilities

1. If the genus of curve $g\gt1$ then by Falting theorem the curve has only finitely many rational points on it .
2. There is also a chance for degree four curves to have genus 1 which falls under the category of Genus-1 curves. So they may have infinite number of solutions and may not have solutions .

But your equation is very vague to start with, you haven't mentioned about any scopes of $X,Y,A$ and also about their meaning. As far as I know we can map the elliptic curve $X^3+Y^3=dZ^3$ and find a corresponding Weierstrass equation by change of variables, if you consider the above curve to be $E$ then we can find a $\phi : E \mapsto \bar E$ by a variable change and then construct the standard equation and also find rational points on them, thats called the "Isogenies".

Once if you clarify it as an elliptic curve, and re-edit the question, I can surely suggest most known methods called Descent and also the Birch Swinnerton-Dyer conjecture, the coefficient gives the better way of constructing points on curve. And there are many such things called " Poonen heuristics and also Prof.M.Stoll methods " , and also Prof.Henri Darmon have proposed many methods in constructing points on curves using $p$-adics

The last thing I want to tell is that if the degree of $y$ is four then they may fall under something called the Hyper-elliptic curves.

So answer depends upon the question. Re-edit it first and be prompt in choosing answers then I can assure that many experts far better than me will help you for sure.

All the best.

Thank you.

Iyengar.