We are tasked with finding all positive integers $n$ such that $n$, $n+2$, and $n+4$ are all prime numbers. Let's analyze this step by step.

Step 1: Explore small values of $n$

First, let's check small values of $n$ to see if we can find any primes that fit the condition.

• For $n = 3$:
  • $n = 3$, $n+2 = 5$, and $n+4 = 7$. All three numbers are prime.
  • Therefore, $n = 3$ is a solution.

Step 2: General Analysis

Now let's see if there are any other solutions.

We are looking for three numbers $n$, $n+2$, and $n+4$ that are all prime. Consider the following points:

• One of the numbers must be divisible by 3, since among any three consecutive numbers $n$, $n+2$, and $n+4$, one of them must be divisible by 3. This follows from the fact that every third number is divisible by 3.

  • If $n$ is divisible by 3, then $n = 3$ (since any higher multiple of 3 is not prime).
  • If $n+2$ is divisible by 3, then $n+2 = 3$, which implies $n = 1$. However, $n = 1$ is not prime, so this case is not possible.
  • If $n+4$ is divisible by 3, then $n+4 = 3$, which implies $n = -1$, but this is not a positive integer.

Thus, the only possibility for $n$ being divisible by 3 is when $n = 3$.

Step 3: Conclusion

The only solution is $n = 3$.

Thus, the only positive integer $n$ such that $n$, $n+2$, and $n+4$ are all prime is: $\boxed{3}$

Would you like further details or clarifications?

Here are five related questions to explore further:

1. What are the first 10 prime numbers?
2. Can there be any sequence of 4 consecutive odd primes?
3. What is the largest known prime number, and how is it discovered?
4. Why is 2 the only even prime number?
5. How can we prove that there are infinitely many prime numbers?

Tip: Prime numbers greater than 3 can be expressed in the form $6k \pm 1$.