The best kinds of ideas about numbers are easy to understand but devilish to actually prove. We saw that in the YouTube channel Numberphile’s video about the Collatz conjecture. Now that channel and the same professor are back to talk about Goldbach’s conjecture.
The conjecture says this: Every even integer greater than 2 can be expressed as the sum of two primes. If you think about small numbers, this is simple. Start with 4—it’s the sum of 2 and 2, and 2 is a prime number. To make 6, you add 3 and 3. To make 8, you can add 3 to 5. And so on.
Things get more interesting as you get higher, because you end up with more ways to add primes to make the same number. For instance, 10 is 3 +7, but it’s also 5+5. You can add 3+97 to make 100, but you can also add 17+83, 11+89, 29+71, 41 +59, and 47+53. Cool, right? It turns out this pattern continues as the target number increases. If you graph that number against the number of possible ways to add two primes, the result is this cool shape called “Goldbach’s comet”. Check out the video for more info, and Numberphile’s YouTube page for bonus interviews.
What’s doubly interesting about Goldbach’s conjecture is that while it’s simple to understand, nobody has been able to prove, mathematically, that this conjecture is true. It sure as hell appears to be true, and no one has even found an example that deviates wildly from the comet trend—that is, a huge even number with very few ways it could be the sum of two primes. Solving this one for good would be a serious mathematician’s coup.
This article about Goldbach’s conjecture was originally written for and published by Popular Mechanics USA.