Problem
NT-B2-M08-P019 A Bounded Solution List
#19
★★★☆☆ Level 3 of 5
Find all positive solutions of \(x^2+y^2+1=3xy\) such that \(x+y\le 150\).
Use the sequence from Problem 8.
By Problem 8, all solutions are neighbouring terms of the sequence \(1,1,2,5,13,34,89,233,\ldots\), where \(a_{n+1}=3a_n-a_{n-1}\). The bound \(x+y\le 150\) leaves the neighbouring pairs \((1,1),(1,2),(2,5),(5,13),(13,34),(34,89)\) and their swaps. The pair \((89,233)\) already has sum greater than \(150\).
A practical reinforcement of the recursive classification.