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A new problem about linear recursive sequence

Source: 2020 China Second Round (A) P3

September 13, 2020
number theoryChinarecurrence relationlinear recurrence

Problem Statement

Let a1=1,a_1=1, a2=2,a_2=2, an=2an−1+an−2,a_n=2a_{n-1}+a_{n-2}, n=3,4,⋯ .n=3,4,\cdots. Prove that for any integer n≥5,n\geq5, ana_n has at least one prime factor p,p, such that p≡1(mod4).p\equiv 1\pmod{4}.
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