# Proving theorems and special cases (Part 6): The Goldbach conjecture

In a recent class with my future secondary math teachers, we had a fascinating discussion concerning how a teacher should respond to the following question from a student:

Is it ever possible to prove a statement or theorem by proving a special case of the statement or theorem?

Usually, the answer is no. In this series of posts, we’ve already seen that a conjecture could be true for the first 40 cases or even the first $10^{316}$ cases yet ultimately prove false for all cases.

For the next few posts, I thought I’d share a few of the most famous unsolved problems in mathematics… and just how much computational work has been done to check for a counterexample.

1. The Goldbach conjecture (see here and here for more information) claims that every even integer greater than 4 can be written as the sum of two prime numbers. For example,

4 = 2 + 2,

6 = 3 + 3,

8 = 3 + 5,

10 = 3 + 7,

12 = 5 + 7,

14 = 3 + 11, etc.

This has been verified for all even numbers less than $4 \times 10^{18} = 4,000,000,000,000,000,000$. A proof for all even numbers, however, has not been found yet.

Here are some results related to the Goldbach conjecture that are known:

1. Any integer greater than 4 is the sum of at most six primes.

2. Every sufficiently large even number can be written as the sum or two primes or the sum of a prime and the product of two primes.

3. Every sufficiently large even number can be written as the sum of two primes and at most 8 powers of 2.

4. Every sufficiently large odd number can be written as the sum of three primes.

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