r^2, s^2, t^2 are all squares, andr^2 + N = s^2 ,s^2+N = t^2 ?

Three Successive Squares w/difference

Here's a Geometric Proof

Three Successive Squares w/difference

Multiplying together square numbers gives us another square number.

Hey,

y^2 = r^2 \cdot s^2 \cdot t^2

y^2 = r^2 \cdot s^2 \cdot t^2 y^2 = (s^2-N) \cdot s^2 \cdot (s^2+N)

y^2 = r^2 \cdot s^2 \cdot t^2 y^2 = (s^2-N) \cdot s^2 \cdot (s^2+N) y^2 = s^6 - N^2 s^2

Next, I want to tell you about a seemingly unrelated problem, which will lead us to an easier way to solve the congruent number problem.

Choose your favorite N , such as 10 . Can you find three square numbers r^2 , s^2 , and t^2 , such that the difference between each two neighboring square numbers is N ?

Fibonacci, the namesake of the Fibonacci numbers, studied this problem. As it turns out, if we can solve this problem and find three square numbers each N apart, it means N is a congruent number! Let's see why.

Let's pretend we already solved the three successive squares problem, and found three square numbers each N apart. Then we can represent these three square numbers geometrically by drawing three squares - one with area r^2 , one with area s^2 , and one with area t^2 .

Because we assumed each square number is N apart, if we remove the square representing r^2 , the area outlined in \text{purple} is 2N . The \text{green area} is N , and the \text{blue L-shaped area} has area N , so combined their area is 2N .

Now, we're going to cut this little rectangle on the bottom...

...and paste it on the side of this shape to make a long rectangle. Since we haven't added or removed any area, this rectangle has the same area as the shape before: its area is 2N .

Finally, drawing a diagonal across this rectangle splits it into two right triangles, with area N !

In other words - if we can solve the three successive squares problem for a certainN , we can follow this recipe to make a triangle with area N , showing N is a congruent number!

In other words - if we can solve the three successive squares problem for a certain

Just to check, we can also calculate the sides of this triangle - and if you do the math, if r , s , and t are rational numbers, we're guaranteed that all the sides are rational numbers. So building this triangle really does show N is a congruent number.

Great - so if we solve the three successive squares problem, we solve the congruent number problem. But how can we find those square numbers and solve the three successive squares problem?

Let's start using some algebra to simplify the three successive squares problem!

Let's start using some algebra to simplify the three successive squares problem!

It's a math fact that if you multiply square numbers together, you get another square number.

If we try to solve the three successive squares problem, we need to find three square numbersr^2 , s^2 , and t^2 . Let's try multiplying r^2 , s^2 , and t^2 together and seeing if we can simplify the equation.

If we try to solve the three successive squares problem, we need to find three square numbers

When we multiply together our three square numbers, r^2 , s^2 , and t^2 , we'll get another square number, which I'll call y^2 .

We can simplify this with some substitution. Our square numbers r^2, s^2, t^2 are all related; once we choose s^2 , we're locked into setting r^2=s^2-N and t^2=s^2+N .

To reduce the number of variables, let's get rid ofr^2 and t^2 by writing them in terms of s^2 : since all three square numbers are N away, we can substitute in r^2 = s^2-N and t^2 = s^2+N .

To reduce the number of variables, let's get rid of

Once we multiply out the terms and simplify, we get the equation y^2 = s^6 - N^2 s^2 . Great - now we've reduced our problem to studying solutions of one equation!

Finally, let's rename s^2 to x , giving us the equation y^2 = x^3 - N^2 x . That's it for the algebra!

Now let's take a look at the equationy^2 = x^3 - N^2 x and see if we can solve it.

Now let's take a look at the equation