A Carry Theorem for Rational Binomial Coefficients

Abstract

Ernst Eduard Kummer proved in 1852 that for any nonnegative integers j and k and any prime p, the exponent of the highest power of p that divides the binomial coefficient $\begin{array}{*{20}{c}} {j + k} k \end{array}$ equals the number of carries that occur when j and k are added together in the p-ary number system. This elegant theorem has been an inspiration and a point of departure for many authors. For example, it has been generalized for i) multinomial coefficients [5], [11], [2]; ii) Gaussian or q-binomial coefficients [3], [4], [6]; iii) “Fibonomial” coefficients [8], [6]; and iv) regularly divisible C-nomial coefficients [6].