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China Mathematical Olympiad 1992 problem6

Source: China Mathematical Olympiad 1992 problem6

September 30, 2013
number theory unsolvednumber theory

Problem Statement

Let sequence {a1,a2,… }\{a_1,a_2,\dots \} with integer terms satisfy the following conditions: 1) an+1=3an−3an−1+an−2,n=2,3,…a_{n+1}=3a_n-3a_{n-1}+a_{n-2}, n=2,3,\dots ; 2) 2a1=a0+a2−22a_1=a_0+a_2-2 ; 3) for arbitrary natural number mm, there exist mm consecutive terms ak,ak−1,…,ak+m−1a_k, a_{k-1}, \dots ,a_{k+m-1} among the sequence such that all such mm terms are perfect squares. Prove that all terms of the sequence {a1,a2,… }\{a_1,a_2,\dots \} are perfect squares.
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