Suppose is a solution.Hello All, in my text it mentions a "method of ascent" procedure for proving that there are infinitely many solutions to certain Diophantine Equations. As an example, it lists:
(x^2) - 3*(y^2) = 1
For familiarity, the book gives the following description for the "method of ascent" :
Method of Ascent shows that given one solution (u,v), another solution can be computed (w,z) which is in a sense larger. The proof will then involve finding a pair of formulas in forms like:
w=x+y and z=x-y. Please note that neither one of thse formulas work.
Specifically for this problem, (x^2) - 3*(y^2) = 1 , it notes that a pair of second degree formulas do work, where one of them has a cross term and the other formula involves the number 3.
Does anyone know how to do this? I'm stumped! Any help is greatly appreciated!
Then and .