# How does the area of a right triangle relate to one of the biggest problems of the millennium? Congruent numbers

My story today begins in a way that the plot is familiar to everyone: with finding the area of a right triangle. I think everyone remembers very well that:

It would seem, what is there to explore, and what does the cutting edge of science, unsolved problems and majestic hypotheses have to do with it? However, mathematics is a deceptive nature: there are very complex problems here ~~sucked out of a finger~~ can grow on the basis of 6th grade formulas, and their rigorous proof can be tied to the truth of one of the 7 mathematical problems of the millennium – the Birch-Swinnerton-Dyer hypothesis. This is exactly what happened in our case!

## Definition

A positive rational number is called **congruent**if it is the area of a right triangle with rational side lengths.

Here a reasonable question arises: “congruent to what?” Usually the term congruent is used as a synonym for the words “equal, identical,” but in our case there are no comparisons. The questions disappear if you look a little into history.

In 1225, King Frederick II of Germany asked Leonardo “Fibonacci” of Pisa to participate in a mathematical tournament, specially for which several problems that were difficult for those times were prepared.

Let’s rewrite this system of equations in one line:

So, the difference of an arithmetic progression, the terms of which are the squares of natural numbers, is called **congruum**. In this case, the congruum is 5. But how to go from an arithmetic progression to a right triangle? Let’s work with the above system of equations in general form:

So what do we see? That’s right, Pythagorean triple! Thus, for an arithmetic progression of three squares **x^2**, **y^2**, **z^2** average member **y** is the hypotenuse of a right triangle, and the other two **x** And **z** – the sum and difference of its legs.

Using the well-known parameterization for Pythagorean triples, we can obtain a general formula that allows us to find such arithmetic progressions and calculate their congruums:

Now let’s calculate the area of the resulting right triangle:

Thus, **60** is a congruent number that matches the congruum **240 **and right triangle **(8,15,17)**.

## Minimum integer congruent number

Pierre Fermat also studied this problem and proved that **1,2 and 3** are not congruent numbers (as the infinite descent method usually used). The smallest congruent integer corresponding to a right triangle with integers **(3,4,5)** the parties are **6**. But there is a possibility that the legs and hypotenuse will not be integers, and the area will be an integer less than **6**?

Let’s return to the example that Fibonacci studied. Let me remind you that in the end he got the following:

Congruum is equal **5**, but what is the area of the corresponding right triangle? Let us solve this problem in general form by finding an expression for the congruum through the parameters **m** And **n**:

Now for a specific case:

We calculate the area and are disappointed:

Area is not a whole number! However, this is not a problem. Let’s see what needs to be done with the sides of the triangle so that its area becomes equal to **5**. It turns out that you can simply double each side of the triangle:

Is this legal? Of course yes! Multiplying by a rational number leaves the three squares as an arithmetic progression, changing only the congruum (in the case of our triangle it became equal to 20).

Since each congruum can be obtained (using parameterization) as the area of a right triangle, **every congruum is a congruent number**. Conversely, every congruent number is the congruum times the square of the rational number.

Please note: to prove that 5 is a congruent number, we used parametrization, solved a non-trivial equation by selection, and then further “corrected” the result. That’s all there is to it: checking that a number is a congruum is easy, but determining its congruence is more difficult. It is here that the question of congruent numbers comes to the forefront of science – algebraic geometry, and more specifically to **theory of elliptic curves.**

## Calculating congruent numbers

By 1915, mathematicians had identified all congruent numbers less than 100. Over the next 65 years, they managed to advance not that far – only up to 1000 (and then with spaces). In 1982, Jerrold Tunnell of Rutgers University made significant progress using the connection between congruent numbers and elliptic curves.

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