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 $\boxed{2,4}$

2006 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
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2006 UNCO Math Contest II Problems/Problem 8

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