2013 IMO Shortlist N3

Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^{4}+n^{2}+1$ is equal to the largest prime divisor of $(n+1)^{4}+(n+1)^{2}+1$. (Belgium)

证明存在无穷多个正整数 $n$，使得 $n^{4}+n^{2}+1$ 的最大素因数等于 $(n+1)^{4}+(n+1)^{2}+1$ 的最大素因数。 (比利时)