# COLLATZ CONJECTURE

Published on
##### Scene 1 (0s)

COLLATZ CONJECTURE

A conjecture is a conclusion made that can be true but due to lack of evidence has not yet been disproved.

##### Scene 2 (2m 2s)

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.

##### Scene 3 (2m 39s)

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

##### Scene 4 (3m 41s)

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.

##### Scene 5 (4m 32s)

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…

##### Scene 6 (5m 3s)

- Consider a positive integer, n n/2 , for even numbers f(n) = 3n – 1 , for odd numbers; Our new sequence will be:

##### Scene 7 (6m 7s)

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.

##### Scene 8 (6m 51s)

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 . ,

##### Scene 9 (7m 44s)

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.

##### Scene 11 (8m 45s)

This conjecture is by far one of the most frustrating to try and solve. Unfortunately we are still yet to prove the conjecture wrong.