Products of binomial coefficients modulo p2

PRODUCTS OF BINOMIAL COEFFICIENTS MODULO p 2

As usual Z, Q, R and C denote the ring of integers, the rational field, the real field and the complex field respectively. We also let Z = {1, 2, 3, · · · } and C∗ = C \ {0}. For a ∈ Z and n ∈ Z, by (a, n) we mean th greatest common divisor of a and n, if n is odd then the Jacobi symbol ( a n ) is defined in terms of Legendre symbols (see, e.g. [IR]). For x ∈ R, [x] and {x} stand for the integr...

A Congruence for Products of Binomial Coefficients modulo a Composite

For a positive composite integer n, we investigate the residues of ( mn k ) for positive integers m and k. First, we discuss divisibility of such coefficients. Then we study congruence identities relating products of binomial coefficients modulo n. Certainly the Chinese Remainder Theorem can be used in combination with prime power results to evaluate binomial coefficients modulo a composite. Ho...

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

On Sums Involving Products of Three Binomial Coefficients

In this paper we mainly employ the Zeilberger algorithm to study congruences for sums of terms involving products of three binomial coefficients. Let p > 3 be a prime. We prove that