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Linear recurrence relations for binomial coefficients modulo a prime

Sandro Mattarei
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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.

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Sandro Mattarei. 2005-11-16. Linear recurrence relations for binomial coefficients modulo a prime. https://doi.org/10.1016/j.jnt.2007.05.003

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