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a finite number of n-tuples

Source: 2012 China TST Test 2 p4

March 20, 2012
inequalitiesinductionnumber theory proposednumber theory

Problem Statement

Given an integer n≥2n\ge 2. Prove that there only exist a finite number of n-tuples of positive integers (a1,a2,…,an)(a_1,a_2,\ldots,a_n) which simultaneously satisfy the following three conditions:
[*] a1>a2>…>ana_1>a_2>\ldots>a_n; [*] gcd⁡(a1,a2,…,an)=1\gcd (a_1,a_2,\ldots,a_n)=1; [*] a1=∑i=1ngcd⁡(ai,ai+1)a_1=\sum_{i=1}^{n}\gcd (a_i,a_{i+1}),where an+1=a1a_{n+1}=a_1.
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