try to prove it with fermats little theorem

To prove that if \( p \) and \( p^2 + 2 \) are prime, then \( p^3 + 2 \) is also prime, we can utilize Fermat's Little Theorem as part of our approach. Fermat's Little Theorem states that if \( p \) is a prime number and \( a \) is any integer not divisible by \( p \), then:

\[ a^{p-1} \equiv 1 \pmod{p} \]

Let's proceed with the given conditions using Fermat's Little Theorem.

### Step 1: Analyze \( p \)
1. If \( p = 2 \):
   \[
   p^2 + 2 = 2^2 + 2 = 4 + 2 = 6
   \]
   Since 6 is not a prime number, \( p \) cannot be 2.

2. Thus, \( p \) must be an odd prime.

### Step 2: Consider \( p^2 + 2 \)
Since \( p \) is an odd prime, let \( p \) be an odd prime number. Given that \( p^2 + 2 \) is prime, \( p \) must be a specific prime value that satisfies this condition.

### Step 3: Use Fermat's Little Theorem
Let \( p \) be a prime. According to Fermat's Little Theorem, for any integer \( a \) not divisible by \( p \):
\[
a^{p-1} \equiv 1 \pmod{p}
\]
We consider \( a = -1 \). Then:
\[
(-1)^{p-1} \equiv 1 \pmod{p}
\]
For odd prime \( p \), \( p-1 \) is even, so:
\[
(-1)^{p-1} = 1
\]
Thus:
\[
1 \equiv 1 \pmod{p}
\]
This shows consistency with Fermat's Little Theorem, but we need further analysis for \( p^3 + 2 \).

### Step 4: Specific Case Analysis
Given the specific condition \( p^2 + 2 \) is prime, let's test small odd primes:
- For \( p = 3 \):
  \[
  p^2 + 2 = 3^2 + 2 = 9 + 2 = 11 \text{ (which is prime)}
  \]
  \[
  p^3 + 2 = 3^3 + 2 = 27 + 2 = 29 \text{ (which is also prime)}
  \]
Thus, \( p = 3 \) satisfies both conditions.

### Step 5: Generalizing the Approach
Now, consider any odd prime \( p \):
1. \( p \) must be such that \( p^2 + 2 \) remains prime.
2. We want to verify \( p^3 + 2 \).

Using the example of \( p = 3 \), this holds true:
\[
p^3 + 2 = 3^3 + 2 = 29 \text{ (which is prime)}
\]

### Conclusion
For \( p = 3 \), \( p \), \( p^2 + 2 \), and \( p^3 + 2 \) are primes, demonstrating \( p^3 + 2 \) can be prime given the initial conditions. Other primes would need similar verification.

Would you like further details or have any questions? Here are 5 possible questions you might ask:
1. Can you explain more about how Fermat's Little Theorem applies here?
2. Are there other examples besides \( p = 3 \) where \( p^2 + 2 \) and \( p^3 + 2 \) are primes?
3. Why did we consider \( a = -1 \) in Fermat's Little Theorem application?
4. How do we rigorously prove the general case for all odd primes?
5. Can Fermat's Little Theorem help in other aspects of number theory proofs?

Tip: Fermat's Little Theorem is useful for simplifying expressions involving primes, especially in modular arithmetic contexts.

Explore how Fermat's Little Theorem is used to prove conditions involving prime numbers. This problem involves verifying if certain expressions like p^3 + 2 remain prime given specific conditions on p and p^2 + 2. Suitable for advanced mathematics enthusiasts.

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