Prove that there exist infinitely many positive integers
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
such that the largest prime divisor of
![n^4 + n^2 + 1](/media/m/e/0/d/e0d088cd148f478320e71d1e846b3b43.png)
is equal to the largest prime divisor of
![(n + 1)^4 + (n + 1)^2 + 1](/media/m/0/1/2/01234b0ddffc2d20d8ed5f940fa74cf0.png)
.
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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 $(n + 1)^4 + (n + 1)^2 + 1$.