Lucas’s theorem


Let m,n-{0} be two natural numbersMathworldPlanetmath . If p is a prime numberMathworldPlanetmath and :

m=akpk+ak-1pk-1++a1p+a0,n=bkpk+bk-1pk-1++b1p+b0

are the base-p expansions of m and n , then the following congruenceMathworldPlanetmathPlanetmathPlanetmathPlanetmath is true :

(mn)(a0b0)(a1b1)(akbk)(𝚖𝚘𝚍𝚙)

Note : the binomial coefficientMathworldPlanetmath is defined in the usual way , namely :

(xy)=x!y!(x-y)!

if xy and 0 otherwise (of course , x and y are natural numbers).

Title Lucas’s theorem
Canonical name LucassTheorem
Date of creation 2013-03-22 13:17:31
Last modified on 2013-03-22 13:17:31
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 6
Author mathcam (2727)
Entry type Theorem
Classification msc 11B65