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Divisibility with recurrence relation

Source: Costa Rica National Olympiad, Final Round, Problem 2

October 14, 2011
number theory unsolvednumber theory

Problem Statement

Consider the sequence xn>0x_n>0 defined with the following recurrence relation:
x1=0x_1 = 0
and for n>1n>1 (n+1)2xn+12+(2n+4)(n+1)xn+1+2n+1+22n−2=9n2xn2+36nxn+32.(n+1)^2x_{n+1}^2 + (2^n+4)(n+1)x_{n+1}+ 2^{n+1}+2^{2n-2} = 9n^2x_n^2+36nx_n+32.
Show that if nn is a prime number larger or equal to 55, then xnx_n is an integer.
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