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-   -   Solving systems of equations modulo n (https://www.mersenneforum.org/showthread.php?t=22009)

carpetpool 2017-02-05 03:03

Solving systems of equations modulo n
 
Like in algebra, how would one solve for x and y in:

4xy = 1 mod 29

2xy^2 = 1 mod 29

Yeah, the solution is obvious and easy in this one, how about when adding or subtracting terms?

x^2-6y = 1 mod 210

3y^2+10y+12x = 1 mod 210

Which one would seem easier to solve?

For both of them, I would go for substitution algebraic method, since the only constant (of degree 0) we are dealing with here in both equations is 1. but I don't know any better. Any help, feedback, or similar equations are appreciated here. Thanks. :smile:

Nick 2017-02-05 09:32

We shall be looking at quadratic equations modulo n shortly in the Basic Number Theory series:
[URL]http://www.mersenneforum.org/forumdisplay.php?f=132[/URL]

CRGreathouse 2017-02-05 20:40

Working mod primes is easy since you have the field structure -- it's just like working over the reals, you can add, subtract, multiply, and divide. Mod prime powers you do much the same thing but then use Hensel lifting. Mod composites you can use the CRT to reduce to prime powers.


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