What mathematical conjecture do you know of that is the easiest to explain, but the hardest to attempt a proof of?

Question

Julia Adams · Accepted Answer

I would say Lothar Collatz's conjecture, which he first proposed in 1937...

Starting with any positive integer #n#, proceed as follows: If #n# is even then divide it by #2#. If #n# is odd, multiply it by #3# and add #1#. The conjecture is that regardless of what positive integer you start with, by repeating these steps you will always eventually reach the value #1#. For example, starting with #7# you get the following sequence: #7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1# If you would like to see a longer sequence, try starting with #27#. This conjecture has been tested for quite large numbers. It looks like it is true, but there is no effective way of solving it with our current mathematical techniques as far as we can tell.

Julia Adams · Answer

undefined One of the mathematical conjectures that fits this description is the Collatz Conjecture. It posits that no matter what positive integer you start with, following a specific rule will always eventually lead to the sequence reaching the number 1. Despite its simplicity, mathematicians have struggled to prove or disprove it for decades.