On the Quadratic Formula Modulo N

Original · EN

Let a, b, c, and n be integers, with a nonzero and n at least two. Necessary and sufficient conditions on these parameters are derived which guarantee that all solutions of the congruence ax²+bx+c ≡ 0modn are given precisely by the solutions of 2ax≡ -b+s modn, where s varies over all solutions of x²≡ b²-4ac modn. Corollaries of this result are deduced for prime-power moduli and some illustrative examples are also presented.