If you are like me, you’ve probably got lots to do for final exams and things of the like as the semester (trimester, quarter, whatever your school uses) comes to a close. In an effort to take a bit of break, I would like to present you with a cute little result I came across in my studies. It’s a nice little number theory fact with a cute combinatorial proof. I found the result in Hatcher’s book on algebraic topology, and I’ll present an adapted version of his proof (though I’m sure it’s in many other books).
First we need to remember a little about -adic representations of integers. If is any integer (technically we should assume , it doesn’t really matter), and if is a prime number, then can be written (uniquely) as , where are integers. This is easy to prove using the fundamental theorem of arithmetic and the division algorithm. Without further adieu I give you the following
Proposition: Let be a prime number and . Then
where and are the -adic representations of and .
Proof: Recall that in the polynomial ring , we have the identity . Working modulo , we have
On one hand, we know that the coefficient of in this expansion is . On the other hand, if we expand the product on the right, it’s easy to see that no two terms in the product will have the same power of . Thus, to compute the coefficient of in the expansion, we need to choose from each factor so that . But we also know that in the ‘th factor, we have terms of the for . In other words, we simply pick so that , which is exactly the -adic representation of . The coefficient on each of these terms if precisely , and so taking the product gives the result.
And that’s it! I hope you enjoyed it, and now I’ve got to get back to work!