Best Rational Approximations to Root 2 generate Pythagorean Triples

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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