In order to find RSA factors ($pq = N$), we have to solve a quadratic equation $x^2+Bx+C=y^2$, where:
- $x_1 < x_2$,
- $x_1, x_2$ and $y_1, y_2$ are (positive) integer numbers,
- $x_1$ is the smallest possible value of $x$ under given conditions,
- we know $B$, $C$, $x_2$ and $y_2$ up front,
- $x_1$ will return us values of $p$ and $q$, while $x_2$ returns us $N$ and $1$.
We are searching for a procedure, that would reveal $x_1$ value. Procedure must not include a factorization of any value, that would result in $p$ and/or $q$ as factors.
Example: $$B=1070637348586356278941580589136$$ $$C=1308129990806270450432723124908$$ $$x_2 = 71641520761751435455133616475131771759770154088777081500862$$
What is the $x_1$ value of the above example? What is the procedure to calculate $x_1$ value (without factorization to $p$ and/or $q$ as a part of procedure)?
Sure, $$x = 22551047929025780409738729886$$
$$x = 71641520761751435455133616475131771759770154088777081500862$$