Question 16

Find all integers $$n$$ such that $$\left|2n^3 - 6n^2 + 4n + 3\right|$$ is prime.


Correct Answer: 0

Solution

PLACEHOLDER ANSWER - the answer is a set of integers. $$2n^3 - 6n^2 + 4n = 2n(n - 1)(n - 2)$$, and the product of three consecutive integers is divisible by 6, so this term is a multiple of 12. The expression is $$12k + 3 = 3(4k + 1)$$, which is prime only when it equals $$\pm 3$$, i.e. $$k = 0$$. So $$n(n - 1)(n - 2) = 0$$, giving $$n = 0, 1, 2$$, where the value is the prime 3.

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