Infinitely prime numbers in a sequence
Source: 35th Austrian Mathematical Olympiad 2004, Round 1, problem 4
February 11, 2009
inductionnumber theoryprime numbersalgebra unsolvedalgebra
Problem Statement
The sequence is defined through:
x_{n \plus{} 1} \equal{} \left(\frac {n}{2004} \plus{} \frac {1}{n}\right)x_n^2 \minus{} \frac {n^3}{2004} \plus{} 1 for
Let be a non-negative integer smaller than so that all members of the sequence are non-negative integers.
Show that there exist infinitely many prime numbers in this sequence.