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Chain of pythagorean triplets

Source: China Mathematical Olympiad 2019 Q2

November 14, 2018
number theoryHi

Problem Statement

Call a set of 3 positive integers {a,b,c}\{a,b,c\} a Pythagorean set if a,b,ca,b,c are the lengths of the 3 sides of a right-angled triangle. Prove that for any 2 Pythagorean sets P,QP,Q, there exists a positive integer m≥2m\ge 2 and Pythagorean sets P1,P2,…,PmP_1,P_2,\ldots ,P_m such that P=P1,Q=PmP=P_1, Q=P_m, and ∀1≤i≤m−1\forall 1\le i\le m-1, Pi∩Pi+1≠∅P_i\cap P_{i+1}\neq \emptyset.
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