using this formula: N = (p1· p2· p3· . . . · pn) + 1 how can N be a non prime number but p1*p2*p3*... must be in sequence and N can only be divided by 2 or 3

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.

Using this formula: N = (p1· p2· p3· . . . · pn) + 1 how can N be a non-prime number but p1*p2*p3*... must be in sequence and N can only be divided by 2 or 3?

Explore how to calculate N using consecutive prime numbers in the formula N = (p1 * p2 * p3 * ... * pn) + 1, ensuring N is a non-prime number divisible only by 2 or 3. This problem involves prime multiplication, divisibility rules, and testing sequences of primes.

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