Method of Ascent

  • Thread starter Frillth
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  • #1
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Homework Statement


I need to prove that the equation x^2 - 3y^2 = 1 has infinite solutions where x and y are both positive integers. I'm supposed to use the method of ascent.

Homework Equations


As a hint, it says to solve this problem by showing how, given one solution (u, v), you can find another solution (w, z) that is larger. Then the proof will involve finding two formulas, like w = x + y and z = x - y. These formulas won't actually work, but there is a pair of second degree formulas which will work. One of them has a cross term and one involves the number 3.

The Attempt at a Solution


The problem is, I've never used the method of ascent before. I have used the method of descent to solve one problem, and I assume that it's like applying descent in reverse. I have no idea how to do this, however. Can I please get some help?
 
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Answers and Replies

  • #2
682
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the famous Pell's equation....
you can find all about it from http://mathworld.wolfram.com/PellEquation.html

hint for the proof:
1. basically, you want to get more solutions from existing ones.

2. suppose you have two solutions to the pell's equation, what happens when you just multiply them (the two equations)?
 

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