This 300-year-old math problem is so difficult there's a $1,000,000 prize to solve it
There exists a math problem so complex and challenging to comprehend that over 300 years and many of the world's most intelligent individuals still can't quite wrap their head around it, even with a $1,000,000 prize dangled in front of them.
While some of the world's most complex problems and hypotheticals are simple to understand in practice, actually proving them objectively can end up being an impossibility that defeats even the brightest minds.
It might be simple enough to assume an answer, or do enough working out to get most of the way, but concretely defining all possible solutions is what allows many of the currently unsolved problems to remain elusive.
With the advancements in technology at our disposal you'd be mistaken for believing that things have become easier, but what's known as 'Goldbach's conjecture' - first proposed all the way back in the 18th century - continues to dominate the minds of many within the math world.
What is Goldbach's conjecture?
In simple terms, Goldbach's conjecture proposes that every even number greater than 2 is the sum of two prime numbers. This means that 4, for example, proves this by being made by adding up 2 and 2, and the same for 8 (5+3), and so on.
Advert
If you're unaware of what prime numbers are, they are values that are only divisible by either 1 or themselves. There are 25 of these between 0 and 100, but this increases exponentially as you go beyond into a potentially infinite amount of prime numbers.
The reason this problem remains so complex and elusive is the sheer possibility of prime numbers far beyond what we can comprehend, and being able to concretely prove that these always add up across two separate instances to create ever single even number is understandably challenging to prove.
It was initially proposed by Prussian mathematician Christian Goldbach in 1742, and split into two separate types: Weak Goldbach Conjecture - which indicates that odd numbers greater than 5 can be created in a sum of three prime numbers - and Strong Goldbach Conjecture, which is the even-number version explained previously.
The odd-number version is named 'weak' as you can create it every single time when you manage to find a Strong version by simply adding another prime number, but that isn't the case for the reverse, hence why it is 'weak' versus 'strong'.
How have people tried to solve Goldbach's conjecture?
When looking at numbers under 100, it's easy to see how the Goldbach conjecture lines up in the creation of a symmetrical pyramid of sorts. YouTube channel Veritasium shows this well in their lengthy video on the topic, as all prime numbers create an intersection of even results when added together, and this, in theory, scales far beyond what you could possibly write or show within a visual like this.
Yet the challenge remains not in 'assuming' that Goldbach's conjecture is always true, but in proving that that's the case, and many began with the weak variant as a starting point.
The 'circle method' is one of the most recognisable efforts towards achieving this, and it uses the function R(n), which measures the number of ways that 'n' can be used in the weak Goldbach conjecture.
This then leads to a separation between what are called 'Major Arcs' and 'Minor Arcs', where the former represents the majority of the contributions laid out in the completed circle in the aforementioned method, whereas the latter is the minority contributions, which are consequently the largest area of the the circle but collect the fewest results.
The Major Arcs are then used as a main term, which measures how many times you can write a number as the sum of three primes (proving Goldbach's weak conjecture) and the Minor Arcs as simply an 'error term'.
Mathematicians then used the Riemann hypothesis to work towards the weak Goldbach conjecture by narrowing the results down to the point where the main terms outweigh the error terms, specifically at the point at which the number can be checked by a computer.
Harald Helfgott managed to achieve this feat alongside David Platt in 2013, concretely proving that the weak Goldbach conjecture is true, yet that's still a long way from achieving the same for the strong.
The closest that anyone has seemingly come to such was Chinese mathmetician Chen Jingrun in 1966, with the aptly named Chen's theorum. This outlined that every 'sufficiently large' even number can be written with either two primes or a prime and a semiprime, although this still doesn't go all the way to indefinitely solve the strong version of Goldbach's original hypothesis.
Veritasium indicate that the easiest way to answer the problem that has troubled mathematicians for over 300 years is to find a number that doesn't fall in line with Goldbach's conjecture, but computers have checked all the way up to 4,000,000,000,000,000,000 (four quintillion) and there's still no value that goes against the hypothesis.
That remains still a fraction of the numbers that need to be checked against the conjecture, but the pattern of how many ways an even number can be written as the sum of two primes (otherwise known as Goldbach's Comet) shows a clean and consistent projection, indicating that its likely to continue this way.
It's unlikely anyone will be able to properly prove or solve the strong Goldbach conjecture any time soon unless a major breakthrough is made with a new theory or a dramatic increase in computational power is gained - even with a $1,000,000 prize as motivation - and only until then will you be able to truly say whether there isn't at least one outlier someone out there.