1. ## Kraitchik method

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

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

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

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

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

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

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

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

Tonio