[nohomo]

How do you find the general integer solution of ?

[oh_1993]

Please note that this is a graphical method and I do not know if it is entirely valid or not.

Consider an x-y graph, the locus of points where x,y are integers are the integer points of the graph i.e. (0,0) (2,3) (1,2) which forms a grid.

Consider the family of curves of the form x^3 + y^3 = c^2, where c is an arbitrary integer constant i.e. x^3 + y^3 = 0 (the origin), x^3 + y^3 = 1, x^3 + y^3 = 4 etc. These are curves for integer values of z

If for a particular value of c, say c=k, the corresponding curve x^3 + y^3 = k^2 passes through an integer point in the x-y plane, say (a,b), then (a,b,k) is an integer solution.

[nohomo]

Thanks for the reply.

I was thinking more of an algebraic way of writing down the general solution, in the same way that any pythagorean triple can be written in the form .

[BabyMaths]

For a start there's (2n^2, 2n^2, 4n^3).

[BabyMaths]

There's also (2n^2, n^2, 3n^3).

[nohomo]

This page give a parametric solution by Euler, but I don't know if it's the general solution, or how to derive it.

[nohomo]

There's also (2n^2,46n^2,312n^3)

[nohomo]

I also found this, but it doesn't give me what I want. I want proof of the general solution.

[BabyMaths]

(Original post by nohomo)
I also found this, but it doesn't give me what I want. I want proof of the general solution.

Look at the bottom of the page of your last link. Apparently there's a proof in this book. http://www.amazon.co.uk/Diophantine-...8234677&sr=1-2

[nohomo]

(Original post by BabyMaths)
Look at the bottom of the page of your last link. Apparently there's a proof in this book. http://www.amazon.co.uk/Diophantine-...8234677&sr=1-2

I had a look at the book on google books and didn't find what I was looking for. Thanks though

[nohomo]

Still struggling with this one.

[nohomo]

What about ?