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 ${\textbf{(E)}\text{ an even integer greater than 2 that cannot be written as the sum of two prime numbers.}}$

Note

Goldbach's conjecture has been proven for all integers less than $4 \cdot 10^{18}$ , but the conjecture remains open. For more information, see the Wikipedia article.

2016 AMC 12A (Problems • Answer Key • Resources)
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All AMC 12 Problems and Solutions

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2016 AMC 12A Problems/Problem 5

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