Reducing multinomial coefficients modulo a prime power

Reciprocal Power Sums Modulo a Prime

Using a sum-product result due to Bourgain, Katz, and Tao, we show that for every 0 < ǫ ≤ 1, and every power k ≥ 1, there exists an integer N = N(ǫ, k), such that for every prime p and every residue class a (mod p), there exist positive integers x1, ..., xN ≤ p ǫ satisfying a ≡ 1 x1 + · · · + 1 xN (mod p). This extends a result of I. Shparlinski [5].

Enumeration of Power Sums Modulo A Prime

We consider, for odd primes p, the function N(p, m, e) which equals the number of subsets SC{1 ,...,p 1) with the property that &sxm = OL (modp). We obtain a closed form expression for N(p, m, IX). We give simple explicit formulas for N(p, 2, CZ) (which in some casea involve class numbers and fundamental units), and show that for a fixed m, the diffenx~ce between N(p, m, a) and its average valu...

Linear Recurrence Relations for Binomial Coefficients modulo a Prime

We investigate when the sequence of binomial coefficients ( k i ) modulo a prime p, for a fixed positive integer k, satisfies a linear recurrence relation of (positive) degree h in the finite range 0 ≤ i ≤ k. In particular, we prove that this cannot occur if 2h ≤ k < p − h. This hypothesis can be weakened to 2h ≤ k < p if we assume, in addition, that the characteristic polynomial of the relatio...

MULTINOMIAL AND g - BINOMIAL COEFFICIENTS MODULO 4 AND MODULO P F * T * Howard

Orders Modulo A Prime

In this article I develop the notion of the order of an element modulo n, and use it to prove the famous n 2 + 1 lemma as well as a generalization to arbitrary cyclotomic polynomials. References used in preparing this article are included in the last page. §1 Introduction I might as well state one of the main results of this article up front, so the following discussion seems a little more moti...