On the Quadratic Formula Modulo N

Steve Wright

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arXiv · 1507.07513

On the Quadratic Formula Modulo N

Steve Wright

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Abstract

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 \[ ax^2+bx+c \equiv 0\ \textrm{mod}\ n \] are given precisely by the solutions of \[ 2ax\equiv -b+s \ \textrm{mod}\ n, \] where $s$ varies over all solutions of \[ x^2\equiv b^2-4ac \ \textrm{mod}\ n. \] Corollaries of this result are deduced for prime-power moduli and some illustrative examples are also presented.

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Steve Wright. 2015-07-27. On the Quadratic Formula Modulo N. https://arxiv.org/abs/1507.07513

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