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221|3^a -2^a, n and n+1 lie in {n\in N^*: 211|1+2^n+3^n+4^n}

Source: China Northern MO 2022 p2 CNMO

October 28, 2022
number theoryDivide

Problem Statement

(1) Find the smallest positive integer aa such that 221∣3a−2a221|3^a -2^a,
(2) Let A={n∈N∗:211∣1+2n+3n+4n}A=\{n\in N^*: 211|1+2^n+3^n+4^n\}.
Are there infinitely many numbers nn such that both nn and n+1n+1 belong to set AA?
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