Look at it in a different way: how many odd numbers CAN be expressed as the sum of two primes? Then subtract that from 50. Clearly one of those primes must be even, and since 2 is the only even prime the odd prime must be two less than the number you're checking. Hence - how may odd prime numbers are there between 3 and 97, inclusive? The answer to how many odds in the range 1 -100 that cannot be expressed as the sum of two primes is 50 minus that number, plus 3 (to account for the numbers 1, 2 and 3). I think you also must include the number 4, despite how they describe the Goldbach Conjecture.
**EDIT - I misread the problem, not realizing that only 2-digit numbers are to be considered. Hence the number is 45 minus the number of primes between 9 and 97, inclusive.
Yes.
Although Goldbach's conjecture has not been proved true generally, it has been verified for all even numbers up to $4 * 10^{18}.$
Thus, you do not need to check any even numbers. All you need to do is to check odd numbers. Obviously, there are 45 odd numbers. But you do not need to check all of them.
Every prime except 2 is an odd number. And the sum of two odd numbers is an even number. So if an odd number is the sum of two primes, one of those primes must be 2. Now consider as an example 2 + 15 = 17. That is a number that cannot be expressed as the sum of two primes because 15 is not a prime. Thus, you only need to consider numbers that are of the form 2 + a prime greater than 7 and less than 100. There are 21 such numbers. So there are 45 - 21 = 24 two-digit numbers that cannot be expressed as the sum of two primes.
Denis needs to go sit in the corner for misreading the question AGAIN.