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The number of elements in the set $$\{n \in \mathbb{N} \mid n^3 - 8n^2 + 20n - 13 \text{ is a prime number}\}$$ is ________
Correct Answer: 3
$$n^3 - 8n^2 + 20n - 13 = (n - 1)(n^2 - 7n + 13)$$, and $$n^2 - 7n + 13 > 0$$ always. For a prime, one factor must be 1. $$n - 1 = 1$$ gives $$n = 2$$ (value 3), and $$n^2 - 7n + 13 = 1$$ gives $$n = 3$$ (value 2) or $$n = 4$$ (value 3). So the set is $$\{2, 3, 4\}$$, with 3 elements.
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