Re: x^3 + y^3 = 22 z^3 how to?

[Elim Qiu <elim.qiu@gmail.com>]

Thanks Bill and Henri, your msgs helped me a lot! 

I'll read GTM 239. 

Best regards,

Elim

On Fri, Aug 9, 2013 at 4:30 PM, <Henri.Cohen@math.u-bordeaux1.fr> wrote:

To complement Bill's message:

1) If (x,y) is a solution then
(-(x^4+2xy^3)/(y^3-x^3),(y^4+2x^3y)/(y^3-x^3))
is also a solution. Note that this is independent of "22".

2) A consequence of the BSD conjecture is that any squarefree
integer congruent to 4,6,7,8 modulo 9 has the same property
(note 22 = 4 modulo 9).

3) This is not an if and only if statement: there exist
certain squaregree integers not congruent to 4,6,7,8 modulo 9
having that property.

For a modern an extensive study of x^3+y^3=nz^3, see my book
Springer GTM 239 Chapter 6, Section 6.4

Henri Cohen