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2016 CNMO Grade 10/11 P7

Source: 2016 China Northern MO, Grade 10, Problem 7; 2016 China Northern MO, Grade 11, Problem 7

February 24, 2020
number theory

Problem Statement

Define sequence (an):an=2n+3n+6n+1(n∈Z+)(a_n):a_n=2^n+3^n+6^n+1(n\in\mathbb{Z}_+). Are there intenger k≥2k\geq2, satisfying that gcd⁡(k,ai)=1\gcd(k,a_i)=1 for all k∈Z+k\in\mathbb{Z}_+? If yes, find the smallest kk. If not, prove this.
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