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Thread: Kraitchik method

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  2. #2
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    First, $\displaystyle 143^2=20,449$, so:

    $\displaystyle u_1=143^2-20,437=12=2^2\cdot 3$

    $\displaystyle u_2=144^2-20,437=299$

    $\displaystyle u_3=145^2-20,437=588=2^2\cdot 3\cdot 7^2$

    We see that $\displaystyle u_1\cdot u_3=2^4\cdot 3^2\cdot 7^2$ , a square, so now:

    $\displaystyle 143\cdot 145=20,735=298\!\!\!\pmod{20,437}\,,\,\,\sqrt{u_1\ cdot u_2}=2^2\cdot 3\cdot 7=84$ , and since $\displaystyle 298\neq 84\!\!\!\pmod{20,437}$, we get:

    $\displaystyle gcd(298-84\,,\,20,437)=107$ , since $\displaystyle 20,437=95\cdot 214+107\,,\,\,214=2\cdot 107$

    Thus, as $\displaystyle \frac{20,437}{107}=191$, we finally get $\displaystyle 20,437=107\cdot 191$

    Tonio
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