Different Ways of Viewing the Goldbach Conjecture
A geometrically flavored approach
When I studied mathematics I often thought about famous unsolved problems before going to sleep. I clearly remember one evening when I was thinking about how to visualize the Goldbach conjecture using circles. Needless to say, I didn’t get much sleep that night!
The Goldbach conjecture is one of mathematics’s oldest and most famous unsolved mysteries.
It started with a casual correspondence between the mathematicians Christian Goldbach and Leonhard Euler in 1742 (even though it was known at least a hundred years before) and has since haunted hopeful young mathematicians and lured them down the mathematical abyss.
In this short article, we will explore this problem by looking at it from different angles. This is sometimes useful for getting new ideas for an attack.
Let us first state the problem as it is usually stated.
The classical description
The Goldbach conjecture states:
For all natural numbers n ≥ 2, there exist two prime numbers p and q such that 2n = p + q.
We sometimes state it as:
“Every even number greater than 2 is the sum of two primes”.
Despite its simple problem statement, this is extremely hard to prove. It is like we need new tools in order to even begin examining it. The criterion has been checked for the first trillions of even numbers by computers but of course, we have infinitely many left to check so we are not very far are we?!
Goldcbach circles
These are the circles mentioned in the introduction.
The idea is the following: If a number 2n can be written as a sum of two primes p and q then these primes must lie equidistant from n on the number line. That is, there must exist an interval with endpoints being primes and center n. This interval should be allowed to be of length 0 if n itself is prime. To…
