Linear recurrence relations for binomial coefficients modulo a prime
Number Theory
2008-04-22 v1
Abstract
We investigate when the sequence of binomial coefficients \binom{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\le i\le k. In particular, we prove that this cannot occur if 2h\le k<p-h. This hypothesis can be weakened to 2h\le k<p if we assume, in addition, that the characteristic polynomial of the relation does not have -1 as a root. We apply our results to recover a known bound for the number of points of a Fermat curve over a finite field.
Cite
@article{arxiv.math/0511417,
title = {Linear recurrence relations for binomial coefficients modulo a prime},
author = {Sandro Mattarei},
journal= {arXiv preprint arXiv:math/0511417},
year = {2008}
}