Linear recurrence relations for binomial coefficients modulo a prime
Sandro Mattarei
Abstract
We investigate when the sequence of binomial coefficients ki 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 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.
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