Define the [i]quasi-primes[/i] as follows.
$\bullet$ The first quasi-prime is $q_1 = 2$
$\bullet$ For $n \ge 2$, the $n^{th}$ quasi-prime $q_n$ is the smallest integer greater than $q_{n_1}$ and not of the form $q_iq_j$ for some $1 \le i \le j \le n - 1$.
Determine, with proof, whether or not $1000$ is a quasi-prime.
To determine whether $1000$ is a quasi-prime, we need to follow the definition of quasi-primes and check if $1000$ can be expressed as a product of two smaller quasi-primes.

1. **Identify the first few quasi-primes:**
   - The first quasi-prime is given: $q_1 = 2$.
   - The second quasi-prime $q_2$ is the smallest integer greater than $2$ and not of the form $q_i q_j$ for $1 \leq i \leq j \leq 1$. Thus, $q_2 = 3$.
   - The third quasi-prime $q_3$ is the smallest integer greater than $3$ and not of the form $q_i q_j$ for $1 \leq i \leq j \leq 2$. Thus, $q_3 = 4$ is not a quasi-prime because $4 = 2 \cdot 2$. The next candidate is $5$, so $q_3 = 5$.
   - The fourth quasi-prime $q_4$ is the smallest integer greater than $5$ and not of the form $q_i q_j$ for $1 \leq i \leq j \leq 3$. Thus, $q_4 = 6$ is not a quasi-prime because $6 = 2 \cdot 3$. The next candidate is $7$, so $q_4 = 7$.
   - The fifth quasi-prime $q_5$ is the smallest integer greater than $7$ and not of the form $q_i q_j$ for $1 \leq i \leq j \leq 4$. Thus, $q_5 = 8$ is not a quasi-prime because $8 = 2 \cdot 4$. The next candidate is $9$, so $q_5 = 9$ is not a quasi-prime because $9 = 3 \cdot 3$. The next candidate is $10$, so $q_5 = 10$ is not a quasi-prime because $10 = 2 \cdot 5$. The next candidate is $11$, so $q_5 = 11$.

2. **Continue identifying quasi-primes:**
   - The sixth quasi-prime $q_6$ is the smallest integer greater than $11$ and not of the form $q_i q_j$ for $1 \leq i \leq j \leq 5$. Thus, $q_6 = 12$ is not a quasi-prime because $12 = 2 \cdot 6$. The next candidate is $13$, so $q_6 = 13$.
   - The seventh quasi-prime $q_7$ is the smallest integer greater than $13$ and not of the form $q_i q_j$ for $1 \leq i \leq j \leq 6$. Thus, $q_7 = 14$ is not a quasi-prime because $14 = 2 \cdot 7$. The next candidate is $15$, so $q_7 = 15$ is not a quasi-prime because $15 = 3 \cdot 5$. The next candidate is $16$, so $q_7 = 16$ is not a quasi-prime because $16 = 4 \cdot 4$. The next candidate is $17$, so $q_7 = 17$.

3. **Check if $1000$ is a quasi-prime:**
   - We need to check if $1000$ can be expressed as a product of two smaller quasi-primes.
   - $1000 = 125 \cdot 8$.
   - From the previous steps, we know that $8$ and $125$ are quasi-primes.

Since $1000$ can be expressed as a product of two smaller quasi-primes ($8$ and $125$), it is not a quasi-prime.

The final answer is $\boxed{\text{No}}$.