10.4. We will call a number greater than 25 semi-prime if it is the sum of some two different prime numbers. What is the maximum number of consecutive natural numbers that can be semi-prime
Answer: 5.

Solution: Note that an odd semiprime number can only be the sum of two and an odd prime number.

We will show that three consecutive odd numbers $2n+1$, $2n+3$, and $2n+5$, greater than 25, cannot all be semiprimes simultaneously. Assuming the contrary, we get that the numbers $2n-1$, $2n+1$, and $2n+3$ are prime, and all are greater than 3. But one of these three numbers is divisible by 3. Contradiction.

Note that among any six consecutive numbers, there are three consecutive odd numbers; therefore, there cannot be more than five consecutive semiprime numbers. Five consecutive numbers can be semiprimes; for example, $30=17+13, 31=29+2, 32=19+13, 33=31+2$, $34=23+11$.

Remark: There are other examples.

Comment: It is proven that there cannot be more than 5 consecutive semiprime numbers (greater than 25) - 4 points.

An example of 5 consecutive semiprime numbers (greater than 25) is provided - 3 points.