I have been trying this diophantine equation
Find all integer solutions to $x^2-xy+y^2=13$
but was unable to crack it.I have solved a few diophantine equations only.So please give a approach to solve this type of many problems and some methods which are used in these equations.
I think that we should try to factorise the equations so that we can pair up the solutions and we can see 13 is a prime number so it is easy to count the solutions or we can create a $(x-y)^2$ to solve and then do the factorisation.
Please while answering try to share the idea one should get while doing these type of questions and the basic approach on should hit to solve.
If there is any trick to solve these equation then please share that also,so that I can use that in further questions.
Thanks in advance
Best Answer
Hint: $x^2-xy+y^2=\frac34(x-y)^2+\frac14(x+y)^2$, so you want to solve $3(x-y)^2+(x+y)^2=52$ for integers $x,y$. Start by bounding $\lvert x-y\rvert$ and $\lvert x+y\rvert$.