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Number Theory

Source: 2000 National High School Mathematics League, Exam Two, Problem 2

March 11, 2020
number theory

Problem Statement

Two sequences (an)(a_n) and (bn)(b_n) satisfy that a0=1,a1=4,a2=49a_0=1,a_1=4,a_2=49, and {an+1=7an+6bn−3bn+1=8an+7bn−4\begin{cases} a_{n+1}=7a_n+6b_n-3\\ b_{n+1}=8a_n+7b_n-4\\ \end{cases} for n=0,1,2,⋯ ,n=0,1,2,\cdots,. Prove that ana_n is a perfect square for n=0,1,2,⋯ ,n=0,1,2,\cdots,.
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