2006 UNCO Math Contest II Problems/Problem 8

Problem

Find all positive integers $n$ such that $n^3-12n^2+40n-29$ is a prime number. For each of your values of $n$ compute this cubic polynomial showing that it is, in fact, a prime.

Solution

Factoring, we get $n^3-12n^2+40n-29 = (n-1)(n^2-11n+29)$. Thus, we must have that either $n-1$ or $n^2-11n+29$ equal to $1$. If we have $n-1$ equal to 1, we have $n=2$. Plugging back in the polynomial, we get $11$, which is a prime, so $2$ works. If $n^2-11n+29$ is equal to one, we have $n^2-11n+28=0$, so $n=4$ or $n=7$. Plugging both back in the polynomial, we get $3$ and $6$, respectively. $3$ is a prime, but $6$ is not, so $4$ works. Thus, the answer is $\boxed{2,4}$

See Also

2006 UNCO Math Contest II (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
1 2 3 4 5 6 7 8 9 10
All UNCO Math Contest Problems and Solutions