1. Representations of the collatz conjecture
2. Final Proof of the Collatz Conjecture

Lother Collatz was a German Mathematician, born 1910- 1990; he proposed the Collatz conjecture in 1937 in which he states that given any positive integer k, the sequence generated by iterations of the Collatz Function will eventually reach and remain in the cycle 4, 2, 1. Collatz states these hailstorm numbers will eventually fall to 1, for any positive interger. The Collatz Conjecture is the most famous unsolved math’s problem. The collatz conjecture is also known as the f(n) = (3n +1 ) /2n problem, the 3n + 1 is the conjecture. If the number is even divide it by 2 until you reach an odd number or 1. If it’s an odd number different from 1 then multiply by 3.

F(n) = { n/2 if n = 0
{ 3n+1 if n = 1

Example 3x+1= 3 +1 = 4 /2 = 2 /2 = 1

Here we have a loop sequence of 4, 2, and 1.

The Collatz Conjecture example below in which n = 10. Because number 10 is even, divide it by 2 and get 5. Now, 5 are odd. So, 3(5) + 1 = 16. 16 is even, so divide by 2 to get 8. Divide 8 by 2 gives 4. Then, 4/2 = 2. 2 divided by 2 is 1. 1 is odd, so ,3(1) +1 = 4. But we already got 4 in sequence, which goes down to 2, then 1. This 4–2–1 loop continues till infinity! So, our finite sequence of unique numbers observed, also called the hailstone sequence is 10, 5, 16, 8, 4, 2, and 1.

The Collatz Conjecture states take any positive integer that is the rule.

An integer is a number which is a whole number. A non-integer is not a whole number. A negative whole number and zero are classed as non-integers. Decimal numbers and fractions are also non integers.

My Proof of the Collatz Conjecture

For two weeks I have been researching the Collatz conjecture, and like most Mathematicians and Scientists have spent many hours trying to prove this unsolved problem. I did many calculations and still I found it came back down to this loop sequence and ending at 1.

My Proof of the Collatz Conjecture

After many hours of study; I began to realise that Mathematicians and Scientists had missed the whole concept of this equation. The Collatz Conjecture is true up until a certain point. After a certain point, it becomes untrue.

Proof:

To prove the collatz conjecture I will consider an example which is
3* 1000000 = 3000000

Apply the Collatz conjecture: Here we have:

3000000/2 = 1,500,000
1,500,000 /2 = 750,000
750,000 /2 = 375,000
375,000 /2 = 187,500
187,500/2 = 93,750
93,750 /2= 46,875
46,875 /2= 23,437.5

This point is where the Collatz conjecture is not true. Because 23,437.5 have converted into a decimal, and the Collatz Conjecture states to use only positive integers, so this is where it ends. It is no longer an integer; it is no longer a whole number.

Let’s just ignore the rule and continue to apply the conjecture with the decimal, as follows:
23,437.5 /2 = 11,718.75

At this point as I see it’s odd add 1 which gives 11,719.75

You see the + 1 does not add onto the .75 leaving it as odd again.

My Proof and Conclusion:

The Collatz Conjecture is true when every Collatz sequence eventually produces 4, 2, 1, 4, 2, and 1. In our case the Collatz Conjecture is true up to the number 46,875, and after these numbers is: 23,437.5, which means we found a, b, c different from 4, 2, and 1 that would disprove the Collatz Conjecture. More generally, if there exists an n-tuple, a_1, a_2, . . . , a_n different from 4, 2, 1, which is eventually infinitely repeated in the sequence of residues, then this would constitute a disproof. Moreover, if we could prove that the only possible n-tuple is 4, 2, and 1 that would only add to the body of evidence supporting the conjecture. The 23,437.5 is a non-Integers and therefore cannot carry on with the Collatz Conjecture. The key to our proof of the Collatz conjecture was that a number must end up in an integer number but this is not true in the Collatz case due to which the Collatz conjecture is false. If the number keeps showing values in the decimal then how can it get down to 1?

The rule is to use only positive integer in the Collatz conjecture. Here you can see there is a decimal point in the 23,437.5 so you cannot possibly carry on the sequence, because decimal point numbers are not allowed to be used in the Collatz conjecture. It seems that Mathematicians and Scientists have gone against the rules of the Collatz Conjecture. As a mathematician I agree that this theory is true because the problem with the Collatz conjecture that makes it so intimately difficult is the unpredictable nature of the cycles. That reason alone is why this problem is so difficult compared to the n+1 problem, which is an incredibly simple inductive proof. Any existing method to prove the Collatz Conjecture using the 3x+1 function directly will fail because the operations are incomplete. But, I hear you say, that the conjecture and the function are the same thing”. I understand this, and I would not contradict that. What I am saying is you need a superset function that is provable and when you have that proof, the smaller case of 3x+1 will have already been proved.

Understanding Mathematical language is the key to understanding the Collatz Conjecture. It is important to observe what the Collatz Conjecture in mathematical terms is asking one to do.
Integers play an important role in the equation. It is clear that Mathematicians have ignored this rule and carried on with calculations up to trillions, whilst including non-integers; which defeat the whole concept of what the Conjecture is asking one to do.

Author: Miss Samiya Hussein is from Lancaster in the UK, and studied Engineering, and she has a very high iQ,

Kenneth

Kenneth is a proud native of sydney, born and raised there. However, he pursued his education abroad and studied in Australia. Kenneth has worked as a journalist for almost a decade, making valuable contributions to prominent publications such as Yahoo News and The Verge. Currently, he serves as a journalist for The Hear Up, where he focuses on covering climate and science news. You can reach Kenneth at [email protected].