Exercise 564Logic, sets and proofs › Quantifiers and negationHigh schoolAdded 3 Nov 2026

Refuting a universal statement

#refutation#counterexample#forall

Statement

Show that $\forall n\in\mathbb{N},\ n^2+n+41$ is prime is false.

Hint

Find $n$ for which the expression is composite.

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Counterexample. At $n=40$ : $n^2+n+41=40^2+40+41=40^2 + 2 \times 40 + 1=41^2$ which is not prime.

Conclusion. The universal statement is false.