# Pythagorean Triangle whose Hypotenuse and Leg differ by 1

## Theorem

Let $P$ be a Pythagorean triangle whose sides correspond to the Pythagorean triple $T$.

Then:

- the hypotenuse of $P$ is $1$ greater than one of its legs

- the generator for $T$ is of the form $G = \tuple {n, n + 1}$ where $n \in \Z_{> 0}$ is a (strictly) positive integer.

## Proof

We have from Solutions of Pythagorean Equation that the set of all primitive Pythagorean triples is generated by:

- $\tuple {2 m n, m^2 - n^2, m^2 + n^2}$

where $m, n \in \Z$ such that:

- $m, n \in \Z_{>0}$ are (strictly) positive integers
- $m \perp n$, that is, $m$ and $n$ are coprime
- $m$ and $n$ are of opposite parity
- $m > n$.

### Necessary Condition

Let the generator $G$ for $T$ be of the form:

- $G = \set {n, n + 1}$

Recall that from Solutions of Pythagorean Equation, $T$ is of the form:

- $\tuple {2 m n, m^2 - n^2, m^2 + n^2}$

where $m, n \in \Z$ such that:

- $m, n \in \Z_{>0}$ are (strictly) positive integers
- $m \perp n$, that is, $m$ and $n$ are coprime
- $m$ and $n$ are of opposite parity
- $m > n$.

We are given that $n \in \Z_{>0}$ such that $m = n + 1$.

We have that Consecutive Integers are Coprime.

They are also of opposite parity.

Therefore the Pythagorean triple generated by $n$ and $n + 1$ is primitive.

So:

\(\displaystyle 2 m n\) | \(=\) | \(\displaystyle 2 n \paren {n + 1}\) | |||||||||||

\(\displaystyle m^2 - n^2\) | \(=\) | \(\displaystyle \paren {n + 1}^2 - n^2\) | |||||||||||

\(\displaystyle m^2 - n^2\) | \(=\) | \(\displaystyle n^2 + 2 n + 1 - n^2\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2 n + 1\) | |||||||||||

\(\displaystyle m^2 + n^2\) | \(=\) | \(\displaystyle n^2 + 2 n + 1 + n^2\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2 n^2 + 2 n + 1\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2 n \paren {n + 1} + 1\) |

Thus it is seen that $P$ is a Pythagorean triangle where:

- one legs is of length $2 n \paren {n + 1}$
- the hypotenuse is of length $2 n \paren {n + 1 + 1$

Thus $P$ is a Pythagorean triangle whose hypotenuse of $P$ is $1$ greater than one of its legs.

$\Box$

### Sufficient Condition

Let the hypotenuse of $P$ be $1$ greater than one of its legs.

Let the legs of $P$ be $a$ and $b$.

Let the hypotenuse of $P$ be $h$ such that $h = b + 1$.

By Consecutive Integers are Coprime, $h$ is coprime to $b$.

Then by Elements of Primitive Pythagorean Triple are Pairwise Coprime, $P$ is primitive.

Because $h$ and $b$ are of opposite parity, it follows that $b$ is even.

Thus by Solutions of Pythagorean Equation:

- $a = m^2 - n^2$
- $b = 2 m n$
- $h = m^2 + n^2$

for some $m, n \in \Z_{>0}$ where $m > n$.

Thus:

\(\displaystyle h\) | \(=\) | \(\displaystyle b + 1\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle m^2 + n^2\) | \(=\) | \(\displaystyle 2 m n + 1\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle m^2 - 2 m n + n^2\) | \(=\) | \(\displaystyle 1\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \paren {m - n}^2\) | \(=\) | \(\displaystyle 1\) | Square of Difference | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle m - n\) | \(=\) | \(\displaystyle 1\) | Definition of Square Root, and we have that $m > n$ | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle m\) | \(=\) | \(\displaystyle n + 1\) |

Hence the result.

$\blacksquare$

## Sequence

The sequence of Pythagorean triangles whose hypotenuse and leg differ by 1 can be tabulated as follows:

- $\begin{array} {r r | r r | r r r} m & n & m^2 & n^2 & 2 m n & m^2 - n^2 & m^2 + n^2 \\ \hline 2 & 1 & 4 & 1 & 4 & 3 & 5 \\ 3 & 2 & 9 & 4 & 12 & 5 & 13 \\ 4 & 3 & 16 & 9 & 24 & 7 & 25 \\ 5 & 4 & 25 & 16 & 40 & 9 & 41 \\ 6 & 5 & 36 & 25 & 60 & 11 & 61 \\ 7 & 6 & 49 & 36 & 84 & 13 & 85 \\ \hline \end{array}$

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $13$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $13$