Prove x^2-3y^2=1 Has Infinite Solutions w/ Method of Ascent

• Frillth
In summary, the conversation is about proving the equation x^2 - 3y^2 = 1 has infinite solutions where x and y are both positive integers using the method of ascent. The person asking for help is given a hint to use a pair of second degree formulas to find new solutions, and is directed to the famous Pell's equation for more information. The conversation also includes a hint for the proof, which involves getting more solutions from existing ones and multiplying two solutions of the Pell's equation.

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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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)?

1. How can the method of ascent be used to prove that x^2-3y^2=1 has infinite solutions?

The method of ascent is a mathematical technique used to prove that a certain equation has an infinite number of solutions. In the case of x^2-3y^2=1, the method involves finding a specific solution to the equation, and then repeatedly adding a certain value to both x and y to generate an infinite number of solutions.

2. What is the specific solution used in the method of ascent for x^2-3y^2=1?

The specific solution for x^2-3y^2=1 used in the method of ascent is (2,1). This means that when x=2 and y=1, the equation is satisfied and equals 1. This solution is used as the starting point to generate an infinite number of solutions.

3. Is the method of ascent the only way to prove that x^2-3y^2=1 has infinite solutions?

No, the method of ascent is not the only way to prove that x^2-3y^2=1 has infinite solutions. Other methods such as proof by contradiction or proof by induction can also be used to show that an equation has infinite solutions.

4. Can the method of ascent be used for other types of equations besides x^2-3y^2=1?

Yes, the method of ascent can be used for other types of equations besides x^2-3y^2=1. It is a general method that can be applied to various types of equations, as long as a specific solution can be found.

5. Are there any limitations to using the method of ascent to prove infinite solutions?

Yes, there are limitations to using the method of ascent to prove infinite solutions. This method can only be used for equations that have a specific solution that can be found and for which a certain value can be added to generate an infinite number of solutions. It may not be applicable to all types of equations.