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\log_{f(1)}f(x) is perfect square if f(x + y)f(x-y) = (f(x)f(y))^2

Source: II - Rioplatense 1992 L3 P1

September 19, 2022
number theoryfunctionalfunctional equation

Problem Statement

Let f:Z→N−{0}f:Z \to N -\{0\} such that:
f(x+y)f(x−y)=(f(x)f(y))2f(x + y)f(x-y) = (f(x)f(y))^2 and f(1)≠1f(1)\ne 1.
Provethat log⁡f(1)f(z)\log_{f(1)}f(z) is a perfect square for every integer zz.
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