The question asks to find the smallest number that can be written as the sum of four distinct primes in two different ways. Let's solve this step by step.

Key Facts:

1. Prime Numbers: We will be using prime numbers (e.g., 2, 3, 5, 7, 11, etc.).
2. Four Primes, Two Different Ways: We are looking for a number that can be written as a sum of four distinct primes in at least two distinct ways.

Step-by-Step Solution:

Step 1: Start with small numbers

We begin with small primes and compute sums of four primes until we find a number satisfying the condition.

Step 2: Check systematically

We start with small combinations of four primes and check whether they sum to the same number for at least two distinct sets of primes.

Calculation:

1. 35 is already given as $2 + 3 + 13 + 17$ and $2 + 3 + 11 + 19$.
   Thus, we skip it and find a smaller example if possible.

2. Trying systematically:

   • Sum = 30:
     Possible combinations of four primes:
     $2 + 3 + 5 + 19 = 29$ (No repeated sums for 30).

     Thus, $30$ does not work.

   • Sum = 34:
     Combinations include:
     $2 + 3 + 7 + 19 = 34$
     $2 + 5 + 11 + 17 = 34$

     This works, as we find two distinct ways to represent 34 as the sum of four primes.

Conclusion:

The smallest number that can be written as the sum of four distinct primes in two different ways is 34.

Would you like me to break this into more steps or explore additional examples? Let me know!

Related Questions:

1. What are all the prime numbers less than 50 that can form sums of four primes in multiple ways?
2. Can this pattern be generalized to larger sets of primes?
3. What is the largest number under 100 that can be written as the sum of four distinct primes in multiple ways?
4. How can we prove that 34 is indeed the smallest such number?
5. How would the solution change if repetition of primes was allowed?

Tip:

When solving problems involving primes, creating a systematic list of combinations can help quickly find patterns or exceptions.