Pascal's triangle revisited

Recall denotes the combinatorial symbol `` choose ''.

If is any prime then , for . (We saw cases of this in Example 1.2.36 above.) This is because divides but does not divide nor ().

Let be the largest power of that divides , where is an integer. The above reasoning tells us that , and for . The exact value of has been known for over 150 years. In 1855, Kummer discovered that is equal to the number of ``carries'' required when adding and in base . For a discussion of this and many other remarkable facts about binomial coefficients, see A. Granville's excellent paper [G].

David Joyner 2007-09-03