The Goldbach Conjecture is one of the most trying problems in all of mathematics. Like the Collatz Conjecture, it is painfully easy to understand, and like the Collatz Conjecture, nobody has been able to prove it. This problem has defied the greatest minds in mathematics, and today it is used as the prototype for problems of this nature.
The Goldbach Conjecture
In 1742, Christian Goldbach proposed a simple idea. He said that every even number can be written as the sum of three primes. Today, since 1 is no longer considered prime, we work with the conjecture that “every even number greater than 2 is the sum of two primes.” This formulation was actually proposed by Euhler in response to the original conjecture, but it is equivalent. (Fourth Grade Refresher Course: prime numbers are the numbers divisible by only 1 and themselves. A few are 2, 3, 5, 7, 11, 13, 17, 19, 23…)
It seems pretty easy on its face. 4 is 2 and 2. 6 is 3 and 3. 8 is 3 and 5. 10 is 5 and 5. 12 is 7 and 5. And so it goes for eternity, at least, we hope.
Proving the Goldbach Conjecture
The Goldbach Conjecture has proven to be a real problem. In spite of its simplicity, there is no method for proving it known today. People have tried for over two hundred and fifty years, and so far, nothing has come close. There are plenty of ways to support the conjecture, but no method of providing a guarantee that it will never fail for a given number.
Of course, disproving it shouldn’t be hard, right? To disprove it, all we need to do is search for a number that doesn’t work. So far, we’re still looking. We’ve gotten up to 4,000,000,000,000,000,000, and every number has worked out to be a sum of primes. Worse yet, if the conjecture is true, this search could go on forever.
The Goldbach Conjecture as a Prototype
The Goldbach Conjecture is used as a prototype for a specific class of problems. These conjectures are “easy” to disprove by means of an exhaustive search, but possibly quite hard to prove. These Goldbach-like problems, such as the Collatz Conjecture , offer up a unique insight into mathematics.
Suppose someone proves that the Goldbach Conjecture or any other Goldbach-like statement is “unprovable” within modern mathematics. This is quite possible thanks to Godel’s Incompleteness Theorem, and it would not be the first time that it happened for a problem. If it were neither provable nor disprovable within our current framework, then that tells you that it is correct. Why? Because if it were incorrect, then it would be totally possible to prove it through that exhaustive search. Thus, proving it to be unprovable is almost akin to proving the exhaustive search will never end, and thus, it is true.
Sources:
Hashref.com: Godel’s Theorem by Xavier Noria
Wolfram Math World: Goldbach Conjecture