Integer both Square and Triangular

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Theorem

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


Then:

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

if and only if:

$\paren {m n}^2$ is both square and triangular.


Proof


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Examples

$\paren {m n}^2$ $m$ $2 n^2 \pm 1 = m^2$ $\dfrac {2 n^2 \paren {2 n^2 \pm 1} } 2$ $m n$ $\dfrac m n$
$1 = 1^2$ $1$ $2 = 2 \times 1^2 - 1$ $\dfrac {2 \times \paren {2 - 1} } 2$ $1 = 1 \times 1$ $1$
$36 = 6^2$ $3$ $9 = 2 \times 2^2 + 1$ $\dfrac {8 \paren {8 + 1} } 2$ $6 = 3 \times 2$ $1 \cdotp 5$
$1225 = 35^2$ $7$ $49 = 2 \times 5^2 - 1$ $\dfrac {50 \paren {50 - 1} } 2$ $35 = 7 \times 5$ $1 \cdotp 4$
$41 \, 616 = 204^2$ $17$ $289 = 2 \times 12^2 + 1$ $\dfrac {288 \paren {288 + 1} } 2$ $204 = 17 \times 12$ $1 \cdotp 41 \dot 6$


Sources

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