Which of the following integers cannot be expressed as the sum of two prime numbers?

Question

Please help me with this problem. I'm stumped!

which of the following integers cannot be expressed as the sum of two prime numbers?

A) $8$

B) $9$

C) $10$

D) $11$

E) $12$

According to the GRE book the answer is.... D) $11$

Answer 1

HINT: $$ 11=1+10=2+9=3+8=4+7=5+6 $$ and there is no other possibility.

Answer 2

Brute force works well for numbers as small as these. Otherwise it helps to know the following facts

• There is no known even number $\ge 4$ that is not a sum of two primes. (And all even numbers with 18 or fewer digits have been tested. See Goldbach's conjecture).

• If an odd number is the sum of two primes, one of them must be $2$. (In other words an odd number $n$ is the sum of two primes if and only if $n-2$ is prime).

Answer 3

The primes less than 12 are 2,3,5,7,11

To get 8:

2+6 (nope); 3 + 5 (yep) 8 = 3+5 is a sum of primes

To get 9:

2+7 = 9. (yep)

To get 10:

2 + 8 (nope); 3 + 7 (yep) (also 5 + 5) 10=3 + 7 = 5 + 5 is the sum of primes

To get 11:

2 + 9 (nope); 3 + 8 (nope); 5 + 6 (nope); 7 + 4 (nope; we've gone past the halfway point; if we were going to find any sum of primes we would have found it already... but lets keep going); 11 + 0; (nope) 11 is not the sum of two primes.

To get 12:

2 + 10(nope); 3 + 9 (nope); 5 + 7 (yep). 12= 5 + 7 is the sum of two primes.