Let us test Euclid's construction on the first few primes. For $L = \{2\}$, $N = 2 + 1 = 3$ (prime); for $L = \{2, 3\}$, $N = 2 \cdot 3 + 1 = 7$ (prime); for $L = \{2, 3, 5\}$, $N = 30 + 1 = 31$ (prime); for $L = \{2, 3, 5, 7\}$, $N = 210 + 1 = 211$ (prime); and for $L = \{2, 3, 5, 7, 11\}$, $N = 2310 + 1 = 2311$ (prime). These first five cases tempt many readers into assuming that $N = p_1 \cdots p_n + 1$ is always prime.

However, at the sixth prime $L = \{2, 3, 5, 7, 11, 13\}$, we compute $N = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 + 1 = 30030 + 1 = 30031$, which is composite because $30031 = 59 \times 509$ (as emphasized in classic texts such as Hardy and Wright's An Introduction to the Theory of Numbers, §1.2). Notice how Euclid's proof in Book IX, Proposition 20 handles this effortlessly: Euclid explicitly splits into the two cases "$EF$ is either prime or not," and when $N = 30031$ is composite, its prime factors $q = 59$ and $q = 509$ are both outside $\{2, 3, 5, 7, 11, 13\}$, giving us not just one but two new primes.