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Prove: For every positive integer $n, n^2 + 4n + 3$ is not a prime.

I tried to disprove the statement, which I could not using several number examples with constructive proof.

However I am not sure how to correctly step by step prove it.

Thank you in advance!

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4 Answers 4

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Hint: $$n^2+ 4n+3=(n+1)(n+3)$$

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  • $\begingroup$ YEs got it! thats basically what I needed! $\endgroup$
    – Nadia S
    Dec 22, 2014 at 16:11
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Hint: $n^2+4n+3 = (n+1)(n+3)$. Can you take it further?

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  • $\begingroup$ yes, thank you! $\endgroup$
    – Nadia S
    Dec 22, 2014 at 16:12
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it is $$(n+1)(n+3)$$ and thus not prime because it has to factors greater than one.

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Recall that $n^2 + 4n + 3 = (n + 1)(n + 3)$. Since $n$ is necessarily a positive integer, it is obvious that $n^2 + 4n + 3$ is composite.

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  • $\begingroup$ What is it that your answer adds to the other three answers? $\endgroup$
    – José Carlos Santos
    Jun 4, 2021 at 18:16

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