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no 6-th power in terms of a_{n+3} = 5a^6_{n+2} + 3a^3_{n+1} + a^2_n

Source: Mathematics Regional Olympiad of Mexico West 2021 P3

September 9, 2022
number theorySequencerecurrence relation

Problem Statement

The sequence of real numbers a1,a2,a3,...a_1, a_2, a_3, ... is defined as follows: a1=2019a_1 = 2019, a2=2020a_2 = 2020, a3=2021a_3 = 2021 and for all n≥1n \ge 1 an+3=5an+26+3an+13+an2.a_{n+3} = 5a^6_{n+2} + 3a^3_{n+1} + a^2_n. Show that this sequence does not contain numbers of the form m6m^6 where mm is a positive integer.
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