MathDB
Good functions with parameter $\alpha$

Source: HMIC 2019 Problem 5

April 28, 2019
algebranumber theoryHMIC

Problem Statement

Let p=2017p = 2017 be a prime and Fp\mathbb{F}_p be the integers modulo pp. A function f:Z→Fpf: \mathbb{Z}\rightarrow\mathbb{F}_p is called good if there is α∈Fp\alpha\in\mathbb{F}_p with α≢0(modp)\alpha\not\equiv 0\pmod{p} such that f(x)f(y)=f(x+y)+αyf(x−y)(modp)f(x)f(y) = f(x + y) + \alpha^y f(x - y)\pmod{p} for all x,y∈Zx, y\in\mathbb{Z}. How many good functions are there that are periodic with minimal period 20162016?
Ashwin Sah
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