$\dfrac 1{344}$ has a recurring decimal period which is $906976744186046511627$.
$\dfrac 1{559}$ has a recurring decimal period which is $178890876565295169946332737030411449016100$
I note that $$\gcd(178890876565295169946332737030411449016100, 906976744186046511627)=\\ 23255813953488372093,$$ which is a big number.
Does other two fractions $\frac{1}{m}$ and $\frac{1}{n}$ for $n$, $m$ integers $m\neq n$ <559 exist such the gcd of their recurring decimal period is greater than $23255813953488372093$?