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x_ax_b is a perfect square

Source: ItaMO 2012, P4

May 21, 2012
modular arithmeticquadraticsnumber theory proposednumber theory

Problem Statement

Let x1,x2,x3,⋯x_1,x_2,x_3, \cdots be a sequence defined by the following recurrence relation: {x1=4xn+1=x1x2x3⋯xn+5 for n≥1 \begin{cases}x_{1}&= 4\\ x_{n+1}&= x_{1}x_{2}x_{3}\cdots x_{n}+5\text{ for }n\ge 1\end{cases} The first few terms of the sequence are x1=4,x2=9,x3=41⋯x_1=4,x_2=9,x_3=41 \cdots
Find all pairs of positive integers {a,b}\{a,b\} such that xaxbx_a x_b is a perfect square.
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