Not to be confused with the The Gold-Bug, the Gold Bug Variations, or the Goldberg Variations, the Goldbach Conjecture is a perhaps second only to Fermat’s Last Theorem among the famous unsolved problems of mathematics. In fact, it’s at the center of at least one book (Uncle Petros and Goldbach’s Conjecture) and one film (Fermat’s Room)
Like many questions in number theory, it’s ridiculously easy to explain to someone over cocktails, but even more ridiculously hard to solve.
What is it?
The better known version is the “strong” Goldbach Conjecture:
Every even integer greater than 2 can be written as the sum of two primes.
The other version is the “weak” Goldbach Conjecture:
Every odd integer greater than 5 can be written as the sum of three primes.
If you think about it for a minute (and subtract 3 🙂 ) you’ll realize that if the “strong” conjecture is true, then the “weak” conjecture is also true. That’s why they’re called “strong” and “weak”, respectively.
From Wikipedia, the history on the problem is thus: On June 7th, 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler in which he told Euler of his conjecture. In reply, Euler wrote
“Dass … ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses theorema, ungeachtet ich dasselbe necht demonstriren kann.” (“every even integer is a sum of two primes. I regard this as a completely certain theorem, although I cannot prove it.”)
In a couple of weeks it will be the 270th anniversary of Goldbach’s Conjecture and it’s still unproven!
Not for lack of trying.
It being the modern era, people have checked it on computers. In particular, Tomás Oliveira e Silva ran a computer search that verified the conjecture for all numbers less than 1,609,000,000,000,000,000 (P.S. He didn’t find a counterexample).
Computer calculations reveal something else: If you make a plot where the x-axis is your number n and the y-axis is the number of ways to write n as a sum of two primes, you might expect the graph to be random looking. However, when you plot the graph you discover that the you actually get a pretty cool looking graph called Goldbach’s comet:
What about on the theoretical side? Various partial results are known (see Wikipedia for details).
In 1995 Ramaré proved that every even integer is the sum of six primes (so a weaker version of the “strong” Goldbach conjecture). In 1995 Kaniecki proved that every odd integer is the sum of at most 5 primes (so a weaker version of the “weak” Goldbach conjecture)*. Unfortunately, Kaniecki’s proof depends on assuming that the Riemann Hypothesis is true. The Riemann Hypothesis is, of course, another equally famous unproven result in mathematics.
Recently, Terry Tao proved that every odd integer is the sum of at most 5 primes. The key thing is that he doesn’t have to assume the validity of the Riemann Hypothesis to do it. So Kaniecki’s theorem is true without any qualifications. It’s a very nice result and Dr. Tao posted about it on his blog.**
* But notice that Kaniecki’s theorem implies Ramaré’s theorem if you subtract by 3.
** It being the Internet, the third comment on Dr. Tao’s blog is a complaint about his choice of title for the paper :-).