2016 AMC 12A Problems/Problem 5

Problem

Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, $2016=13+2003$). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?

$\textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\ \qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\ \qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\ \qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\ \qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}$

Solution

In this case, a counterexample is a number that would prove Goldbach's conjecture false. The conjecture asserts what can be done with even integers greater than 2. Therefore the solution is $${\textbf{(E)}\text{ an even integer greater than 2 that cannot be written as the sum of two prime numbers.}}$$