The master Paul Erdos said "Mathematical may not be ready for such problem"
Terence Tao recently proposed a new and advanced approach for this conjecture and concluded: "Almost all orbits of the Collatz map attain almost bounded values".
The Collatz 's conjecture is infamous and very hard to solve
Take any positive integer, if it is even divide it by 2. If it is odd , multiply the number with 3 and add 1. Whatever the answer , repeat the same operations on the result.
Suppose the number is 5 then the operations wil be as follows: 5, 16, 8, 4, 2,1,4,2,1
Suppose the number is 7 then the operations will be as follows:7,22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1,4,2,1
The conjecture has been verified by computer for number as big as 10^18 and respects all the powers of 2. This is easely checked: 128, 64, 32, 16, 8, 4, 2, 1, 4, 2, 1.
How any positive integer reach some power of 2 in order to reach the loop of 4, 2, 1.?
We claim that any positive integer has a special numbber equal to a multiple of the positive integer. When the operation of 3n+1 is performed on that multiple it leads to some power of 2 .
N=1 gives special multiple 5=5*1.
3*5+1=16=2^4
N=3 gives special multiple 21=3*7
3*21+1=64=2^6
N=5 gives special multiple 85=17*5
3*85+1=256=2^8
The set (1, 5, 21, 85, 341.....) are called Collatz Numbers.
So we can claim that the Collatz's conjecture is almost solved.
Dear Mohamed Azzedine
You succeed to generate a set that follows the Collatz conjecture.
We can generate many similar examples by reversing the mentioned procedure,
but this approach can't be accepted as proof.
Regards
Please read "any odd and positive integer has a special number equal to a multiple of the positive integer." instead of
"any positive integer has a special number equal to a multiple of the positive integer."
The Collatz number (1, 5, 21,85 ,341, 1365,.5461, 21845...) most of them are multiple of 5, more than 50%.
This characteritic is shown in an excellent paper written by Mr. Sparsh Joshi.
"94% of the number reached 5 before becoming a power of two".
.
Dear Issam Kaddoura
Thank you for your comment .
Could you provide a similar example of Collatz number for any odd integer?
Best regards
Dear Mohamed Azzedine
Consider the following family ( your answer)
{(22m+2- 4)/3+1,m = 0,1,2,3,....} = {1, 5, 21, 85, 341, 1365, 5461,.....}
Another family is
{(7x22m- 4)/3 + 1, m = 1, 2, 3,....} = { 9, 37, 149, 341,597, ....}
Following Collatz"s approach, all end by 7 and hence agree with Collatz's conjecture.
Regards
Dear Issam Kaddoura
The first family is the family I already proposed and called it Collatz number. All the elements are power of two and converge quickly to the loop 4, 2, 1.
The second family are composed of element which are not power of two nor 7 and converge slowly to 7. It is fully different from Collatz conjecture. It is irrelevant.
The set (1,5, 21, 341,1365, 5161, 21845...) is a break through and an advance made in the Collatz conjecture.
Wait and see.
Best regards
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