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 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.
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 . 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.