COLLATZ CONJECTURE

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COLLATZ CONJECTURE

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

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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.

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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

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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.

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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…

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

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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.

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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 . ,

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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.

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