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Putnam 2019 A5

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December 10, 2019
Putnam 2019Putnamnumber theory

Problem Statement

Let pp be an odd prime number, and let Fp\mathbb{F}_p denote the field of integers modulo pp. Let Fp[x]\mathbb{F}_p[x] be the ring of polynomials over Fp\mathbb{F}_p, and let q(x)∈Fp[x]q(x) \in \mathbb{F}_p[x] be given by q(x)=∑k=1p−1akxkq(x) = \sum_{k=1}^{p-1} a_k x^k where ak=k(p−1)/2a_k = k^{(p-1)/2} mod pp. Find the greatest nonnegative integer nn such that (x−1)n(x-1)^n divides q(x)q(x) in Fp[x]\mathbb{F}_p[x].
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