A natural number is called a [i]prime power[/i] if that number can be expressed as $p^n$ for some prime $p$ and natural number $n$. 
Determine the largest possible $n$ such that there exists a sequence of prime powers $a_1, a_2, \dots, a_n$ such that $a_i = a_{i - 1} + a_{i - 2}$ for all $3 \le i \le n$. 
To determine the largest possible \( n \) such that there exists a sequence of prime powers \( a_1, a_2, \dots, a_n \) satisfying \( a_i = a_{i-1} + a_{i-2} \) for all \( 3 \le i \le n \), we will analyze the given cases and prove by contradiction that \( n \geq 8 \) is not possible.

1. **Case 1: \( a_1 \) and \( a_2 \) are both even**

   Suppose \( a_1 = 2^x \) and \( a_2 = 2^y \). Then:
   \[
   a_3 = a_1 + a_2 = 2^x + 2^y
   \]
   Without loss of generality, assume \( x \geq y \). Then:
   \[
   a_3 = 2^y (2^{x-y} + 1)
   \]
   For \( a_3 \) to be a prime power, \( 2^{x-y} + 1 \) must be a power of 2. However, the only power of 2 that is one more than another power of 2 is 2 itself (i.e., \( 2^1 + 1 = 3 \)). This implies \( x = y + 1 \), and:
   \[
   a_3 = 3 \cdot 2^y
   \]
   Since \( a_3 \) must be a prime power, this is a contradiction because \( 3 \cdot 2^y \) is not a power of a single prime.

2. **Case 2: \( a_1 \) and \( a_2 \) are both odd**

   Suppose \( a_1 \) and \( a_2 \) are both odd prime powers. Then:
   \[
   a_3 = a_1 + a_2
   \]
   Since the sum of two odd numbers is even, \( a_3 \) must be even. Let \( a_3 = 2^z \). Then:
   \[
   a_4 = a_2 + a_3 = a_2 + 2^z
   \]
   Since \( a_2 \) is odd and \( 2^z \) is even, \( a_4 \) is odd. Continuing this pattern, we find that:
   \[
   a_6 = 3a_1 + 5a_2
   \]
   Since \( a_1 \) and \( a_2 \) are both odd, \( 3a_1 + 5a_2 \) is also odd, which contradicts the requirement that \( a_6 \) be a prime power.

3. **Case 3: \( a_1 \) is even and \( a_2 \) is odd**

   Suppose \( a_1 = 2^x \) and \( a_2 \) is an odd prime power. Then:
   \[
   a_3 = a_1 + a_2 = 2^x + a_2
   \]
   Since \( a_2 \) is odd, \( a_3 \) is odd. Continuing this pattern, we find that:
   \[
   a_7 = 5a_1 + 8a_2
   \]
   Since \( 5 \) and \( a_2 \) are both odd, \( 5a_1 + 8a_2 \) must be even. However, for \( a_7 \) to be a prime power, it must be a power of 2, which contradicts the requirement that \( a_7 \) be a prime power.

4. **Case 4: \( a_1 \) is odd and \( a_2 \) is even**

   Suppose \( a_1 \) is an odd prime power and \( a_2 = 2^y \). Then:
   \[
   a_3 = a_1 + a_2
   \]
   Since \( a_1 \) is odd and \( a_2 \) is even, \( a_3 \) is odd. Continuing this pattern, we find that:
   \[
   a_6 = 3a_1 + 5a_2
   \]
   Since \( a_1 \) is odd and \( a_2 \) is even, \( 3a_1 + 5a_2 \) must be odd, which contradicts the requirement that \( a_6 \) be a prime power.

Since all four cases lead to contradictions when \( n \geq 8 \), we conclude that the largest possible \( n \) is 7.

The final answer is \( \boxed{7} \).