Unfamiliar notation in number theory

dafunk

Came across the notation of ^{-1} with respect to modulos in a friends coursework yesterday, did a lot of number theory back in the day but it was a long long time ago now and am very out of practice so it could be something really obvious, it said

Find 12 * 18^{-1} modulo 23

Presumably the power of minus 1 is not referring to a recipricol as we don't use fractions with respect to congruence modulos.

Eliot Rosewater

Probably means the inverse of 18, which you'd find using Euclid's algorithm.

Michael Collins

Yeh it's possibly an unLaTeX-ed snippet of LaTeX code. If I were to write that in LaTeX I'd get:

[latex] \displaystyle 12 \times 18^{-1} \hbox{ modulo }23[/latex].

CJC86

As Elliot Rosewater noted, it's the inverse of 18, which means it is [latex] 18^{-1} = x \in \{ 0,\ldots , 22\}[/latex] such that [latex]x.18=1 \mbox{mod}23[/latex].

(I said x in {0,...,22} for illustrative purposes. Strictly speaking, it would be the equivalence class mod23, which x in {0,...,22} is a representative of)