Special Isosceles Triangles

Consider the isosceles triangle with base length, $b = 16$, and legs, $L = 17$.

By using the Pythagorean theorem it can be seen that the height of the triangle, $h = \sqrt{17^2 – 8^2} = 15$, which is one less than the base length.
With $b = 272$ and $L = 305$, we get $h = 273$, which is one more than the base length, and this is the second smallest isosceles triangle with the property that $h = b \pm 1$.
Find $\sum L$ for the twelve smallest isosceles triangles for which $h = b \pm 1$ and $b$, $L$ are positive integers.

This is a fairly complex problem that involves some understanding of number theory and Pythagorean triples.

For an isosceles triangle with base $b$ and legs $L$, we have $h^2 + (b/2)^2 = L^2$ due to the Pythagorean theorem, where $h$ is the height of the triangle. Given that $h = b \pm 1$, we can substitute this into the equation and get $(b \pm 1)^2 + b^2/4 = L^2$, or equivalently, $5b^2/4 \pm b + 1 = L^2$.

Since $L$ must be an integer, $b$ must be divisible by 4, meaning $b = 4n$ for some integer $n$. After substitution, we have $5n^2 \pm n + 1 = L^2$.

Let’s transfer it to $L^2 – 5n^2 = \pm n – 1$, which is a Pell equation and then we can use its property of adding consecutive solutions.

Here’s how to solve the Pell equation:

We can let $L + \sqrt{5} n = (9 + 4\sqrt{5})(2 + \sqrt{5})^k$, so when $k = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11$, we get the $x$ when the sequence starts at $k=0$. Then each consecutive term in the sequence is formed as $x_{n+1} = 6x_n – x_{n-1}$, with initial terms $x_0 = 2$ and $x_1 = 9$.

For each $x_n$, we can get $y_n = (x_n^2 – 1)/5$. If $x_n$ is odd, then $n = x_n – 1$, and if $x_n$ is even, then $n = x_n + 1$. Keeping only the cases where $n$ is divisible by 4 will give us the desired solutions, $b = n$ and $L = y_n$.

Only considering $n = 4m$ where $x_n$ is odd if $n$ is odd and $x_n$ is even if $n$ is even, and considering 12 smallest $L$, you need to manually check that $(b, L)$ gives $h = b \pm 1$ until we get 12 such $L$ and take the sum.

So you can solve the problem by generating the sequence of $x_n$, calculating $y_n$, then checking whether it meets the conditions outlined above until we have 12 values of $L$. Then we sum these values of $L$ together to get the answer. Note that the details of sequence generation, Pell equation solutions and sequence properties can be involved and might require a further understanding of number theory.

More Answers: