The Collatz conjecture is one of the most famous unsolved mathematical
problems, because it's so simple, you can explain it to a primary-schoolaged kid, and they'll probably be intrigued enough to try and find the
answer for themselves.
So here's how it goes: pick a number, any number.
If it's even, divide it by 2. If it's odd, multiply it by 3 and add 1. Now repeat
those steps again with your new number. Eventually, if you keep going,
you'll eventually end up at 1 every single time (try it for yourself, we'll wait).
As simple as it sounds, it actually works. But the problem is that even
though mathematicians have shown this is the case with millions of
numbers, they haven't found any numbers out there that won't stick to the
rules.
"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," explains Thompson. "But no one has ever been able to prove that for
certain."
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
but
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.