Area of a Triangle and Elliptic Curves - Birch and Swinnerton Dyer Conjecture

• Math Amateur
In summary, in Chapter 6 of Keith Devlin's book on the Millennium Problems, it is stated that there is a right triangle with rational sides having an area d if and only if the equation y^2 = x^3 - d^2x has rational solutions for x and y with y ≠ 0. This can be shown through straightforward algebraic reasoning, which is explained in detail in section 3 of the book. The term "congruence number" is also mentioned as a key concept in this proof.
Math Amateur
Gold Member
MHB
In the book by Keith Devlin on the Millenium Problems - in Chapter 6 on the Birch and Swinnerton-Dyer Conjecture we find the following text:

"It is a fairly straightforward piece of algebraic reasoning to show that there is a right triangle with rational sides having an area d if and only if the equation:

$y^2 = x^3 - d^2 x$

has rational solutions for x and y with

(Note: Devlin has defined d as a positive whole number earlier on)

Can someone please supply the straightforward algebra to show that:

There is a right triangle with rational sides having an area d if and only if the equation:

$y^2 = x^3 - d^2 x$

has rational solutions for x and y with $y \ne 0$"

Peter

I think it's a bit disingenuous to say that this is straightforward. Sure, if someone gives you the substitutions then you can execute them with no difficulty. But coming up with these substitutions on your own is not so trivial (in my opinion).

Anyway, the key phrase here is "congruence number". See section 3 here for the details you're after.

1. What is the formula for finding the area of a triangle?

The formula for finding the area of a triangle is A = 1/2 * base * height, where A represents the area, base represents the length of the triangle's base, and height represents the height of the triangle.

2. What is an elliptic curve?

An elliptic curve is a type of mathematical curve that has a specific equation in the form of y^2 = x^3 + ax + b. These curves have a variety of applications in number theory, cryptography, and other fields.

3. What is the Birch and Swinnerton-Dyer Conjecture?

The Birch and Swinnerton-Dyer Conjecture is a mathematical conjecture that relates the number of rational points on an elliptic curve to the behavior of its L-function (a type of mathematical function). It is considered one of the most important unsolved problems in mathematics.

4. Why is the Birch and Swinnerton-Dyer Conjecture important?

The Birch and Swinnerton-Dyer Conjecture has important implications in number theory and cryptography. It also has connections to other areas of mathematics, such as algebraic geometry and complex analysis.

5. Has the Birch and Swinnerton-Dyer Conjecture been proven?

No, the conjecture has not been proven. It remains an open problem in mathematics, and many mathematicians continue to work on finding a proof or counterexample.

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