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Parities aren't periodic

Source: 2023 Grosman MO, P5

September 8, 2023
number theorySequenceParity

Problem Statement

Consider the sequence of natural numbers ana_n defined as a0=4a_0=4 and an+1=an(an−1)2a_{n+1}=\frac{a_n(a_n-1)}{2} for each n≥0n\geq 0. Define a new sequence bnb_n as follows: bn=0b_n=0 if ana_n is even, and bn=1b_n=1 if ana_n is odd. Prove that for each natural mm, the sequence bm,bm+1,bm+2,bm+3,…b_m, b_{m+1}, b_{m+2},b_{m+3}, \dots is not periodic.
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