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Modular periodicity of binomial coefficients

✍ Scribed by Sandro Mattarei

Publisher: Elsevier Science
Year: 2006
Tongue: English
Weight: 160 KB
Volume: 117
Category: Article
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2005.07.005

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✦ Synopsis

We prove that if the signed binomial coefficient (-1) i k i viewed modulo p is a periodic function of i with period h in the range 0 i k, then k + 1 is a power of p, provided h is not too large compared to k. (In particular, 2h k suffices). As an application, we prove that if G and H are multiplicative subgroups of a finite field, with H < G, and such that 1 -∈ G for all ∈ G\H , then G ∪ {0} is a subfield.

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