8. (10 points) The largest odd number that cannot be written as the sum of three distinct composite numbers is

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8. (10 points) The largest odd number that cannot be written as the sum of three distinct composite numbers is
【Analysis】Among the positive integers, the three smallest composite numbers are $4, 6, 8$. First, calculate their sum, then compare it with the nearest odd number, and finally prove the correctness of this conclusion.

【Solution】Solution: Among the positive integers, the three smallest composite numbers are $4, 6, 8$. Their sum is $4+6+8=18$, so 17 is an odd number that cannot be expressed as the sum of three different composite numbers. Next, we will prove that any odd number $n$ greater than or equal to 19 can be expressed as the sum of three different composite numbers.

Since when $k \geqslant 3$, $4, 9, 2k$ are three different composite numbers, and $4+9+2k \geqslant 19$, by appropriately choosing $k$, any odd number $n$ greater than or equal to 19 can be expressed as the sum of $4, 9, 2k\left(k=\frac{n-13}{2}\right)$. In summary, the largest odd number that cannot be expressed as the sum of three different composite numbers is 17. Therefore, the answer is: 17.