2012-10-12, 06:42 | #1 |
Apr 2011
7 Posts |
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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