MathDB
JBMO Shortlist 2022 N4

Source: JBMO Shortlist 2022

June 26, 2023
number theorySequencerecursiveJuniorBalkanshortlist

Problem Statement

Consider the sequence u0,u1,u2,...u_0, u_1, u_2, ... defined by u0=0,u1=1,u_0 = 0, u_1 = 1, and un=6un−1+7un−2u_n = 6u_{n - 1} + 7u_{n - 2} for n≥2n \ge 2. Show that there are no non-negative integers a,b,c,na, b, c, n such that ab(a+b)(a2+ab+b2)=c2022+42=un.ab(a + b)(a^2 + ab + b^2) = c^{2022} + 42 = u_n.
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