Problem:
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, ). 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?
Answer Choices:
A. an odd integer greater than 2 that can be written as the sum of two prime numbers
B. an odd integer greater than 2 that cannot be written as the sum of two prime numbers
C. an even integer greater than 2 that can be written as the sum of two numbers that are not prime
D. an even integer greater than 2 that can be written as the sum of two prime numbers
E. an even integer greater than 2 that cannot be written as the sum of two prime numbers
Solution:
A counterexample must satisfy the hypothesis of being an even integer greater than 2 but fail to satisfy the conclusion that it can be written as the sum of two prime numbers.
The problems on this page are the property of the MAA's American Mathematics Competitions