Prime power divisors of binomial coefficients

Pacific Journal of Mathematics, 1969

A classical theorem discovered independently by J. Sylvester and I. Schur states that in a set of k consecutive integers, each of which is greater than k 9 there is a number having a prime divisor greater than k. In giving an elementary proof, P. Erdόs expressed the theorem in the following form:

Journal of The London Mathematical Society-second Series, 1974

has shown that the density of binomial coefficients divisible by a prime p is one. Rosch 5], on the basis of numerical evidence, has conjectured that the density of binomial coefficients divisible by any integer d is one. In this note, I shall prove this in the following form. THEOREM 1. For any positive integer d, almost all binomial coefficients are divisible byd. "Almost all" can have many meanings. I shall give proofs for four different meanings, which appear to be the most appropriate. In any case, it is clear that if we have d = Yl Pi e *, then it suffices to show that each p%H divides almost every coefficient, so we may and do restrict our attention to d = p e .

We present a general formula for the number of binomial coefficients in a given row of Pascal’s triangle that are divisible by p^j and not divisible by p^{j+1}, where p is a prime.

The American Mathematical Monthly, 2018

We define a new class of integer partitions generalizing hyperbinary partitions. We then utilize these partitions to provide a new, relatively simple formula for the number of binomial coefficients in a particular row of Pascal's triangle that are divisible by a fixed power of a prime. Our formula specializes to a famous result due to N. J. Fine from 1947. Furthermore, we demonstrate a connection between the binomial coefficients congruent to zero modulo a fixed prime power and the so-called digital dominance orders on the natural numbers. A classic problem in combinatorial number theory asks, for a fixed prime number p and a given integer n, how many binomial coefficients of the form n k are divisible by p. We visualize this problem by shading Pascal's triangle so that an entry n k is unshaded (white) if it is not divisible by p and it is shaded (blue) if it is divisible by p. For instance, the upper left image in Figure shows the picture for p = 2. Kung [10] appears to be the first author to include this type of image as a method of visualizing Pascal's triangle modulo a prime. This question naturally extends to divisibility by a fixed prime power, p m , in Pascal's triangle. The remaining three images in Figure show Pascal's triangle reduced modulo 4, 8, and 16, again where we shade the entries divisible by the appropriate power of 2. Of course, the corresponding problem arising from these diagrams is to find a formula for the number of shaded entries in a given row. As luck would have it, the literature abounds with details about and formulas for this quantity. Fine [7] utilizes Lucas's theorem to enumerate the number of entries in row n of Pascal's triangle that are nonzero modulo p. Carlitz [3] gives recursive formulas and generating functions for these counts when p m is fixed, and Everett [5] gives a closed form for the functions introduced by Carlitz in terms of binary words. Furthermore, Rowland and Everett [6] give alternative interesting and efficient formulas for computing the number of entries in row n of Pascal's triangle congruent to 0 mod p m . Finally, Spiegelhofer and Wallner have recently answered a variety of questions posed by Rowland [13]; the results in those papers can be seen as generalizations of Fine's result. In the current note, we add another chapter to this saga by providing a formula for the number of entries congruent to 0 mod p m in row n of Pascal's triangle that uses a certain family of integer partitions. Our connection between Pascal's triangle, these partitions, and the formula for the number of zero entries appears to have been missed in the vast literature about Pascal's triangle mod p m and requires only a simple, relatively nontechnical proof. Moreover, our results can be interpreted in terms of a family of partial orders defined on the natural numbers, which are known as base-p digital dominance orders (see ).

We use the notion of subprime factorization to establish recurrence relations for the number of binomial coefficients in a given row of Pascal’s triangle that are divisible by p^j and not divisible by p^{j+1}, where p is a prime. Using these relations to compute this number can provide significant savings in the number of computational steps.