Sum of Primes

Since $100001$ is odd, then one of the two primes must be even while the other is odd. Clearly one of them is $2$ and so the other is $100001-2=9999$ . But $9999$ is a composite number, hence, it can never be written as the sum of two primes.

Because 100,001 is an odd number, one number must be even and other must be odd. The only even prime number is 2, thus the other prime must be 99,999.

Clearly 99,999 is divisible, thus there can be no ways 100,001 can be written as the sum of two primes.

Answer = 0