A conjecture is a conclusion made that can be true but due to lack of evidence has not yet been disproved.
Lothar Collatz suggests , Considering a positive integer , n n/2 , for an even number f(n) = 3n + 1 , for an odd number Regardless of what number you choose, If you follow the algorithm, we get a hailstone sequence which eventually reaches 1.
E.g. 5: 5→ 16→ 8→ 4→ 2→ 1 12: 12→ 6→ 3→ 10→ 5→ 16→ 8→ 4→ 2→ 1 Using a computer, This is the case for all positive integers less than 2 68
It can be a very tedious job looking for that one number that does not follow the above sequence and as such we are yet to discover an argument against Collatz Conjecture.
However, mathematicians have tried coming up with loop holes where the conjecture could fail. For example, looking for a cycle that gives a Collatz-like sequence that doesn’t result in the known cycle 1,4,2,1…
- Consider a positive integer, n n/2 , for even numbers f(n) = 3n – 1 , for odd numbers; Our new sequence will be:
For n = 7: 7→ 20→ 10→ 5→ 14→ 7→… n = 17: 17→ 50→ 25→ 74→ 37→ 110→ 55→ 164→ 82→ 41→ 122→ 61→ 192→ 91→ 272→ 136→ 68→ 34→ 17→… Although it doesn’t reach 1 we still cannot use this as our evidence.
Terence Tao’s work has come the closest at trying to rebut Collatz. He applied knowledge on partial differential equations to show the existence of upper bounds of the Col (n) b y using logarithmic density on its beta invariant properties . ,
He also used the technique of weights choices to look for a sample of numbers that mostly retain its original weights all through the Collatz process that will give a value very close to 1.
This conjecture is by far one of the most frustrating to try and solve. Unfortunately we are still yet to prove the conjecture wrong.