We all know that maths is really hard. So hard, in fact, that there’s literally a whole Wikipedia page dedicated to unsolved mathematical problems, despite some of the greatest minds in the world working on them around the clock.
But as Avery Thompson points out at Popular Mechanics, from the outset at least, some of these problems seem surprisingly simple – so simple, in fact, that anyone with some basic maths knowledge can understand them… including us. Unfortunately, it turns out that proving them is a little harder.
Pick any number. If that number is even, divide it by 2. If it’s odd, multiply it by 3 and add 1. Now repeat the process with your new number. If you keep going, you’ll eventually end up at 1. Every time.
Mathematicians have tried millions of numbers and they’ve never found a single one that didn’t end up at 1 eventually. The thing is, they’ve never been able to prove that there isn’t a special number out there that never leads to 1. It’s possible that there’s some really big number that goes to infinity instead, or maybe a number that gets stuck in a loop and never reaches 1. But no one has ever been able to prove that for certain.
The Twin Prime conjecture
Prime numbers are those magical unicorns that are only divisible by themselves and 1. As far as we know, there’s an infinite number of primes, and mathematicians are working hard to constantly find the next largest prime number.
But is there an infinite amount of prime numbers pairs that differ by two, like 41 and 43? As primes get larger and larger, these twin primes are harder to find, but in theory, they should be infinite… the problem is no one’s been able to prove that as yet.
The Beal conjecture
The Beal conjecture basically goes like this…
If Ax + By = Cz
And A, B, C, x, y, and z are all positive integers (whole numbers greater than 0), then A, B, and C should all have a common prime factor.
A common prime factor means that each of the numbers needs to be divisible by the same prime number. So 15, 10, and 5 all have a common prime factor of 5 (they’re all divisible by the prime number 5).
So far, so simple, and it looks like something you would have solved in high school algebra.
But here’s the problem. Mathematicians haven’t ever been able to solve the Beale conjecture, with x, y, and z all being greater than 2.
For example, let’s use our numbers with the common prime factor of 5 from before….
51 + 101 = 151
52 + 102 ≠ 152
There’s currently a US$1 million prize on offer for anyone who can offer a peer-reviewed proof of this conjecture… so get calculating.
Similar to the Twin Prime conjecture, Goldbach’s conjecture is another seemingly simple question about primes and is famous for how deceptively easy it is. The question here is: is every even number greater than 2 the sum of two primes?
It sounds obvious that the answer would be yes, after all, 3 + 1 = 4, 5 + 1 = 6 and so on.
But, again, no one’s been able to prove that this will always be the case, despite years of trying.