MathDB

4

Part of 2012 ITAMO

Problems(1)

x_ax_b is a perfect square

Source: ItaMO 2012, P4

5/21/2012
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.
modular arithmeticquadraticsnumber theory proposednumber theory