Difference between revisions of "Polynomial congruences"

Revision as of 21:57, 18 January 2025

Polynomial Congruences are congruences in the form

$f(x) \equiv 0 \pmod {m}$

where $f(x)$ is an arithmetic function and a polynomial whose range is the integers and $m$ is an integer.

There are a few ways of solving polynomial congruences, and special cases can make it easier.

Some polynomial congruences can be solved obviously. For example, the congruence

$16x^2 \equiv 13 pmod {192}$

obviously have no solution, since $\gcd (16,192)$ don't divide $13$ .

Additionally, if $f(x)$ is of

@@ Line 3: / Line 3: @@
 <cmath>f(x) \equiv 0 \pmod {m}</cmath>
-where <math>f(x)</math> is an [[arithmetic function]] whose range is the integers and <math>m</math> is an integer.
+where <math>f(x)</math> is an [[arithmetic function]] and a [[polynomial]] whose range is the integers and <math>m</math> is an integer.
-To solve polynomial congruences, we use a result called [[Hensel's Lemma]].
+==Solving==
+There are a few ways of solving polynomial congruences, and special cases can make it easier.
+===Introductary===
+Some polynomial congruences can be solved obviously. For example, the congruence
+<cmath>16x^2 \equiv 13 pmod {192}</cmath>
+obviously have no solution, since <math>\gcd (16,192)</math> don't divide <math>13</math>.
+Additionally, if <math>f(x)</math> is of
 ==See Also==
@@ Line 12: / Line 22: @@
 {{stub}}
-[[Category:Definition]]
+[[Category:Definitions]]