To satisfy the conditions in the problem where $N$ must be a non-prime number, and $N = (p_1 \cdot p_2 \cdot p_3 \cdot \ldots \cdot p_n) + 1$, let's break down the steps:

1. The sequence $p_1, p_2, \ldots, p_n$ must consist of consecutive prime numbers.
2. $N$ must not be a prime number, meaning it should have divisors other than 1 and itself.
3. The divisors of $N$ can only be 2 or 3, so $N$ must be divisible by either 2 or 3.

Example Process:

Step 1: Choose a sequence of consecutive primes

Start with small prime numbers for the sequence $p_1, p_2, p_3, \ldots$.

For example, let’s take the first three primes: $p_1 = 2, \quad p_2 = 3, \quad p_3 = 5$

Step 2: Calculate $N$

Using the formula: $N = (p_1 \cdot p_2 \cdot p_3) + 1 = (2 \cdot 3 \cdot 5) + 1 = 30 + 1 = 31$ In this case, $N = 31$, which is a prime number, so this doesn't work.

Step 3: Try different sequences

Now try four consecutive primes: $p_1 = 2, \quad p_2 = 3, \quad p_3 = 5, \quad p_4 = 7$ $N = (2 \cdot 3 \cdot 5 \cdot 7) + 1 = 210 + 1 = 211$ Again, 211 is a prime number, so it doesn't satisfy the condition.

Step 4: Continue expanding the sequence

Now, try five consecutive primes: $p_1 = 2, \quad p_2 = 3, \quad p_3 = 5, \quad p_4 = 7, \quad p_5 = 11$ $N = (2 \cdot 3 \cdot 5 \cdot 7 \cdot 11) + 1 = 2310 + 1 = 2311$ 2311 is also a prime number.

Step 5: Adjusting for the divisibility rule

To find $N$ that satisfies the condition of being divisible by 2 or 3, we need to experiment further with sequences of primes and verify the conditions.

Final Answer:

The approach is to test sequences of primes and verify if $N$ satisfies the conditions. You can continue expanding the prime sequence until $N$ becomes divisible by 2 or 3 and is no longer prime. However, there's no straightforward pattern for such sequences; it requires checking larger products.

Would you like to see further calculations with larger primes, or do you have any questions about the process?

5 Related Questions:

1. How do prime numbers behave in sequences when multiplied?
2. Can a formula like this always generate a prime number?
3. How can one prove that a number is divisible only by 2 or 3?
4. What methods exist for checking the primality of a large number?
5. Why does adding 1 to the product of primes often result in a prime number?

Tip:

When dealing with sequences of prime numbers and checking for divisibility, using a computer algorithm can help quickly test large sequences to see which satisfy the conditions.