Kummer’s theorem
Given integers and a prime number , then the power of dividing is equal to the number of carries when adding and in base .
Proof.
For the proof we can allow of numbers in base with leading zeros. So let
all in base . We set and denote the -adic representation of with .
We define , and for each
(1) |
Finally, we introduce as the sum of digits in the -adic of . Then it follows that the power of dividing is
For each , we have
Then
This gives us
the total number of carries. ∎
Title | Kummer’s theorem |
---|---|
Canonical name | KummersTheorem |
Date of creation | 2013-03-22 13:22:37 |
Last modified on | 2013-03-22 13:22:37 |
Owner | Thomas Heye (1234) |
Last modified by | Thomas Heye (1234) |
Numerical id | 14 |
Author | Thomas Heye (1234) |
Entry type | Theorem |
Classification | msc 11A63 |