Generator for Almost Isosceles Pythagorean Triangle
Theorem
Let $P$ be a Pythagorean triangle whose sides correspond to the Pythagorean triple $T = \tuple {a, b, c}$.
Let the generator for $T$ be $\tuple {m, n}$.
Then:
- $P$ is almost isosceles
- $\tuple {2 m + n, m}$ is the generator for the Pythagorean triple $T'$ of another almost isosceles Pythagorean triangle $P'$.
Proof
By definition of almost isosceles:
- $\size {a - b} = 1$
First note that, from Consecutive Integers are Coprime, an almost isosceles Pythagorean triangle is a primitive Pythagorean triangle.
Hence $T$ and $T'$ are primitive Pythagorean triples.
Thus it is established that by Solutions of Pythagorean Equation, both $P$ and $P'$ are of the form:
- $\tuple {2 m n, m^2 + n^2, m^2 - n^2}$
for some $m, n \in \Z_{>0}$ where:
- $m > n$
- $m \perp n$
- $m$ and $n$ are of opposite parity.
Necessary Condition
Let $\tuple {2 m + n, m}$ be the generator for the Pythagorean triple $T'$ of the almost isosceles Pythagorean triangle $P'$.
Let $p$ and $q$ be the legs of $P'$.
By Solutions of Pythagorean Equation:
- $p = 2 \paren {2 m + n} m$
- $q = \paren {2 m + n}^2 - m^2$
Thus:
\(\ds \size {2 \paren {2 m + n} m - \paren {\paren {2 m + n}^2 - m^2} }\) | \(=\) | \(\ds 1\) | Definition of Almost Isosceles Pythagorean Triangle | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \size {4 m^2 + 2 m n - \paren {4 m^2 + 4 m n + n^2 - m^2} }\) | \(=\) | \(\ds 1\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \size {\paren {m^2 - n^2} - 2 m n}\) | \(=\) | \(\ds 1\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \size {a - b}\) | \(=\) | \(\ds 1\) | Solutions of Pythagorean Equation |
That is, $P$ is almost isosceles.
$\Box$
Sufficient Condition
Let $P$ be almost isosceles.
Without loss of generality, by Solutions of Pythagorean Equation:
- $a = 2 m n$
- $b = m^2 - n^2$
Thus:
\(\ds \size {a - b}\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \size {\paren {m^2 - n^2} - 2 m n}\) | \(=\) | \(\ds 1\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \size {4 m^2 + 2 m n - \paren {4 m^2 + 4 m n + n^2 - m^2} }\) | \(=\) | \(\ds 1\) | working the above sequence of equations in reverse | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \size {2 \paren {2 m + n} m - \paren {\paren {2 m + n}^2 - m^2} }\) | \(=\) | \(\ds 1\) | Solutions of Pythagorean Equation |
By Solutions of Pythagorean Equation, $2 \paren {2 m + n} m$ and $\paren {2 m + n}^2 - m^2$ are the legs of a Pythagorean triangle $P'$ whose generator is $\paren {2 m + n, m}$.
But from the above, these legs differ by $1$.
Hence, by definition, $P'$ is an almost isosceles Pythagorean triangle $P'$.
$\blacksquare$
Sequence
The sequence of almost isosceles Pythagorean triangles can be tabulated as follows:
- $\begin{array} {r r | r r | r r r}
m & n & m^2 & n^2 & 2 m n & m^2 - n^2 & m^2 + n^2 \\ \hline 2 & 1 & 4 & 1 & 4 & 3 & 5 \\ 5 & 2 & 25 & 4 & 20 & 21 & 29 \\ 12 & 5 & 144 & 25 & 120 & 119 & 169 \\ 29 & 12 & 841 & 144 & 696 & 697 & 985 \\ 70 & 29 & 4900 & 841 & 4060 & 4059 & 5741 \\ 169 & 70 & 28 \, 561 & 4900 & 23 \, 660 & 23 \, 661 & 33 \, 461 \\ \hline \end{array}$
The sequence of elements of the generators are the Pell numbers:
- $1, 2, 5, 12, 29, 70, 169, 408, \ldots$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $13$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $13$