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Two sequences

Source: Iberoamerican Olympiad 1992, Problem 4

May 14, 2007
number theory proposednumber theory

Problem Statement

Let {an}n≥0\{a_{n}\}_{n \geq 0} and {bn}n≥0\{b_{n}\}_{n \geq 0} be two sequences of integer numbers such that: i. a0=0a_{0}=0, b0=8b_{0}=8. ii. For every n≥0n \geq 0, an+2=2an+1−an+2a_{n+2}=2a_{n+1}-a_{n}+2, bn+2=2bn+1−bnb_{n+2}=2b_{n+1}-b_{n}. iii. an2+bn2a_{n}^{2}+b_{n}^{2} is a perfect square for every n≥0n \geq 0. Find at least two values of the pair (a1992, b1992)(a_{1992},\, b_{1992}).
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