Question 169428: I really do not understand proofs.
I was given this question:
Goldbach's conjecture is that every integer greater than 2 can be expressed as the sum of two primes. What would a counterexample for this conjecture be? (or what characteristics would a couterexample have?)
Thank You
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website! First of all, you have misstated Goldbach's conjecture. The conjecture, properly stated, is:
Every even integer greater than 2 can be expressed as the sum of two primes.
First thing is to define 'counterexample' A counterexample is an example that disproves the conjecture. A simple illustration would be thus: Let's say that you came to me with the conjecture that "All dogs are black." I could then disprove your conjecture by finding a white dog (or a brown one, for that matter).
A counterexample to the Goldbach conjecture would be to find an even integer (the counterexample number) where every prime number less than the counterexample number when subtracted from the counterexample number results in a non-prime difference.
So, if  (even and greater than 2) is the counterexample number, and {  , , ,..., } is the set of ALL prime numbers less than , then NONE of the subtractions  (where i = 1 through p) results in a prime number.
The strong Goldbach conjecture has never been proven in general, but neither has a counterexample ever been found. If you want to go off on your own search for an actual counterexample to Goldbach, start with numbers greater than  because every possibility less than that has already been shown to fit the conjecture.
|
|
|
