1. **Problem Statement:** We are given the recursive definition of Euclidean numbers: $$e_1 = 2$$ $$e_n = 1 + \prod_{k=1}^{n-1} e_k \quad \text{for } n > 1$$ We want to understand the first few Euclidean numbers and verify their properties. 2. **Understanding the formula:** The formula states that each Euclidean number $e_n$ is 1 plus the product of all previous Euclidean numbers $e_1, e_2, ..., e_{n-1}$. 3. **Calculate the first four Euclidean numbers:** - $e_1 = 2$ (given) - $e_2 = 1 + e_1 = 1 + 2 = 3$ - $e_3 = 1 + e_1 \times e_2 = 1 + 2 \times 3 = 1 + 6 = 7$ - $e_4 = 1 + e_1 \times e_2 \times e_3 = 1 + 2 \times 3 \times 7 = 1 + 42 = 43$ 4. **Check the primality of these numbers:** - $2, 3, 7, 43$ are all prime numbers. 5. **Calculate $e_5$:** $$e_5 = 1 + e_1 \times e_2 \times e_3 \times e_4 = 1 + 2 \times 3 \times 7 \times 43 = 1 + 1806 = 1807$$ 6. **Factorize $e_5$:** $1807 = 13 \times 139$, so $e_5$ is not prime. 7. **Property:** Euclidean numbers are pairwise prime, meaning any two distinct Euclidean numbers share no common prime factors. **Final answer:** The first four Euclidean numbers are $2, 3, 7, 43$, all prime. The fifth Euclidean number is $1807$, which is composite ($13 \times 139$).