# Best Rational Approximations to Root 2 generate Pythagorean Triples

## Theorem

Consider the Sequence of Best Rational Approximations to Square Root of 2:

- $\left\langle{S}\right\rangle := \dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \dfrac {41} {29}, \dfrac {99} {70}, \dfrac {239} {169}, \dfrac {577} {408}, \ldots$

Every other term of $\left\langle{S}\right\rangle$ can be expressed as:

- $\dfrac {2 a + 1} b$

such that:

- $a^2 + \left({a + 1}\right)^2 = b^2$
- $b$ is odd.

## Proof

From Parity of Best Rational Approximations to Root 2:

- The numerators of the terms of $\left\langle{S}\right\rangle$ are all odd.

- For all $n$, the parity of the denominator of term $S_n$ is the same as the parity of $n$.

Thus it follows that every other term of $\left\langle{S}\right\rangle$ has a numerator and a denominator which are both odd.

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $1 \cdotp 41421 \, 35623 \, 73095 \, 04880 \, 16887 \, 24209 \, 69807 \, 85697 \ldots$