Integer both Square and Triangular

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Theorem

Consider a pair of (strictly) positive integers $a$ and $b$ such that $a < b$.


Then:

$\dfrac a b$ is a best rational approximation to the square root of $2$

if and only if:

$\paren {a b}^2$ is both square and triangular.


Proof


Examples

$n$ $\dfrac {k \paren {k + 1} } 2$ $m^2$ $a b = m$ $\dfrac a b$
$1$ $\dfrac {1 \paren {1 + 1} } 2$ $1^2$ $1 \times 1$ $1$
$36$ $\dfrac {8 \paren {8 + 1} } 2$ $6^2$ $3 \times 2$ $1 \cdotp 5$
$1225$ $\dfrac {49 \paren {49 + 1} } 2$ $35^2$ $7 \times 5$ $1 \cdotp 4$
$41 \, 616$ $\dfrac {288 \paren {288 + 1} } 2$ $204^2$ $17 \times 12$ $1 \cdotp 41 \dot 6$


Sources