Problem

Source: Romania IMO TST 1989 1.1

Tags: Sequence, number theory, prime



Let the sequence ($a_n$) be defined by $a_n = n^6 +5n^4 -12n^2 -36, n \ge 2$. (a) Prove that any prime number divides some term in this sequence. (b) Prove that there is a positive integer not dividing any term in the sequence. (c) Determine the least $n \ge 2$ for which $1989 | a_n$.