Question

What attempts had been made when people tried to prove the Collatz Conjecture?

What approaches were there?

Answer 1

A few thoughts...

Explanation:

The great Polish mathematician Paul Erdős said of the Collatz conjecture that "Mathematics may not be ready for such problems.". He offered a $500 prize for a solution.

It seems as intractable today as when he said that.

It is possible to express the Collatz problem in several different ways, but there is no real method to try to solve it. When I was at university nearly 40 years ago the only idea people seemed to have was to look at it using 2-adic arithmetic.

I thought of trying to address it using some kind of measure-theoretical approach, but about the best that could do would probably be to show that the set of numbers which do not hit `1` is of measure `0`. It would not exclude the existence of counterexamples.

The Collatz conjecture has been checked by computer for numbers up to about `10^20`, but that only really shows that it is plausible - it does not prove it to be true for all numbers.

To understand why iterative processes such as that in the Collatz conjecture are so hard to solve in general, it may help to see how rich the combination of addition and multiplication on natural numbers actually is.

For example, if you define any formal mathematical system with a finite number of symbols and allowed operations, then basic arithmetic is sufficient to codify it. It then becomes possible to construct an algebraic statement which interpreted says effectively "I am not provable in this formal system". Such a statement is then true but not provable. So the formal system is provably incomplete.

This is roughly the essence of the proof of Gödel's second incompleteness theorem.