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-   -   residues and non residues of general quadratic congruences (https://www.mersenneforum.org/showthread.php?t=17288)

smslca 2012-10-12 06:42

residues and non residues of general quadratic congruences
 
for a given range of x in Zn , and n is composite , and ax² + bx + c ≡ 0(mod n) and if (4a,n)=1,
I learned that we can solve the congruence by (2ax + b)² ≡ b²-4ac (mod n) ==> y² ≡ z (mod n)

but my question is , how many z values will be QR mod n and how many will be NQR and what about their distribution in Zn . And how many could be solved and how many could not be solved in the range of x values using y² ≡ z (mod n).? and can we predict x value for given z value , if x is some subset of Zn but not Zn

I could only find literatures on n being a prime, but what if n is composite?

For example , let n = 38411 and 10<x<√(n) or some bounds, i.e some arbitrary length of range of x >0

So , can any one suggest , where can i find literature on my doubt . or even books or any means i can solve my problem.
can anyone guide me in wright and exact path , i should take.

If i am wrong or obscure any where in my question , hope will be notified to me.


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