We are tasked with finding all positive integers $n$ such that $n$, $n+2$, and $n+4$ are all prime numbers.

Step 1: Consider small values of $n$

We start by checking small values of $n$, since these numbers need to be primes, and prime numbers tend to grow sparse as $n$ increases.

• For $n = 3$:

  • $n = 3$ (prime),
  • $n+2 = 5$ (prime),
  • $n+4 = 7$ (prime).

  So, $n = 3$ works.

• For $n = 5$:

  • $n = 5$ (prime),
  • $n+2 = 7$ (prime),
  • $n+4 = 9$ (not prime).

  So, $n = 5$ does not work.

• For $n = 7$:

  • $n = 7$ (prime),
  • $n+2 = 9$ (not prime),
  • $n+4 = 11$ (prime).

  So, $n = 7$ does not work.

• For $n = 11$:

  • $n = 11$ (prime),
  • $n+2 = 13$ (prime),
  • $n+4 = 15$ (not prime).

  So, $n = 11$ does not work.

• For $n = 13$:

  • $n = 13$ (prime),
  • $n+2 = 15$ (not prime),
  • $n+4 = 17$ (prime).

  So, $n = 13$ does not work.

Step 2: General analysis

Let’s now analyze the problem more generally. We want $n$, $n+2$, and $n+4$ to all be prime numbers.

• Modulo 3 reasoning: Since any integer $n$ is congruent to either 0, 1, or 2 modulo 3, we will check each case to see if it leads to a contradiction.

  • If $n \equiv 0 \mod 3$, then $n$ is divisible by 3. The only prime divisible by 3 is 3 itself. Hence, $n = 3$ is a solution.

  • If $n \equiv 1 \mod 3$, then $n+2 \equiv 0 \mod 3$, meaning $n+2$ is divisible by 3. Since the only prime divisible by 3 is 3 itself, we must have $n+2 = 3$, which leads to $n = 1$. But $n = 1$ is not prime, so there is no solution in this case.

  • If $n \equiv 2 \mod 3$, then $n+4 \equiv 0 \mod 3$, meaning $n+4$ is divisible by 3. As before, $n+4$ must be 3, which gives $n = -1$, but $n$ must be a positive integer. Hence, there is no solution in this case.

Conclusion

The only solution is $n = 3$, where $n = 3$, $n+2 = 5$, and $n+4 = 7$ are all prime numbers.

Would you like more details or have any questions?

Here are 5 related questions to further explore the topic:

1. What are twin primes, and how are they related to this problem?
2. How sparse are prime triplets as $n$ increases?
3. Are there other prime triplets of the form $n, n+2, n+4$?
4. How can modular arithmetic help in solving problems involving primes?
5. Why do primes become less frequent as numbers get larger?

Tip: When solving problems involving primes, checking small cases first can give helpful insights into patterns or constraints that apply to larger cases.