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Fake NT \gcd(a_1, a_2, \dots, a_n)

Source: INAMO 2020 P8

October 14, 2020
number theoryconstructionBoundingbait

Problem Statement

Determine the smallest natural number n>2n > 2, or show that no such natural numbers nn exists, that satisfy the following condition: There exists natural numbers a1,a2,…,ana_1, a_2, \dots, a_n such that gcd⁡(a1,a2,…,an)=∑k=1n−1(1gcd⁡(ak,ak+1)+1gcd⁡(ak,ak+2)+⋯+1gcd⁡(ak,an))⏟n−k terms \gcd(a_1, a_2, \dots, a_n) = \sum_{k = 1}^{n - 1} \underbrace{\left( \frac{1}{\gcd(a_k, a_{k + 1})} + \frac{1}{\gcd(a_k, a_{k + 2})} + \dots + \frac{1}{\gcd(a_k, a_n)} \right)}_{n - k \ \text{terms}}
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