mathschallenge.net

Problem

We shall define an $almost equilateral triangles$ to be a triangle for which two sides are equal and the third differs by no more than one unit. The smallest such triangle with integral length sides and area is 5-5-6.

Prove that infintitely many $almost equilateral triangles$ with integral length sides and area exist.

Solution

By the definition an $almost equilateral triangles$ measuring $a-a-b$ is isosceles.

Using the Pythagorean Theorem: $a$² = ($b$/2)² + $h$², so 4$a$² = $b$² + 4$h$².

As $b$ = $a$ 1, $b$² = $a$² 2$a$ + 1

4$a$² = $a$² 2$a$ + 1 + 4$h$²
    3$a$² 2$a$ 1 + 4$h$² = 0
    9$a$² 6$a$ 3 12$h$² = 0
    9$a$² 6$a$ + 1 12$h$² = 4
    (3$a$ 1)² 12$h$² = 4
((3$a$ 1)/2)² 3$h$² = 1

By writing $x$ = (3$a$ 1)/2 and $y$ = $h$, we get the Pell equation: $x$² 3$y$² = 1. Given one solution, it is well known that Pell equations have infinitely many solutions, and for completeness we shall prove this.

However, we must first show that integer $x$ corresponds to an integer solutions for $a$; $b$ being integer follows as $b$ = $a$ 1.

As $x$ = (3$a$ 1)/2, we get $a$ = (2$x$1)/3

It should be clear that $x$ cannot be divisible by 3, otherwise $x$² 3$y$² would be a multiple of 3 and could not be equal to 1.

So given that $x$ 1 mod 3, 2$x$ 1 mod 3, and so one of 2$x$+1 or 2$x$1 will be a multiple of 3. Hence for every integer solution of the equation $x$² 3$y$² = 1, we have an integer solution for $a$ and $b$. Now we shall prove that infinitely many solutions exist.

Given ($x$,$y$), a solution pair to the Pell equation $x$² 3$y$² = 1, consider the larger pair ($x$²+3$y$²,2$xy$):

($x$2+3$y$2)2 3(2$xy$)2	=	$x$4 + 6$x$2$y$2 + 9$y$4 12$x$2$y$2
	=	$x$4 6$x$2$y$2 + 9$y$4
	=	($x$2 3$y$2)2
	=	1

In other words, if ($x$,$y$) is a solution then ($x$²+3$y$²,2$xy$) will also be a solution, and as (7,4) leads to the first solution 5-5-6, we prove that infinitely many $almost equilateral triangles$ with integral length sides and area exist.

Note that although the infinite solution set of the Pell equation $x$² 3$y$² = 1 is in a one-to-one mapping with the set of $almost equilateral triangles$ we are seeking, this particular iterative method: ($x$,$y$) ($x$²+3$y$²,2$xy$), will NOT produce every solution.