Proof of Collatz Theorem Pure & Simple Way!
Proof of Collatz Theorem
Pure & Simple
Every Number That is equalized to the power of 2 Reaches To
One (GOD)
3n+b ==> an + b ==>
(2x+1)m + 2y +1 = 2^k ,
Collatz
Theorem : Any positive integer divide
by 2 if it's even, and multiply by 3 if it's odd and add 1, you'll always get
1.
Collatz's
Proof: When the odd number in the Collatz set is equalized to the power of 2, that is, when an + b = 2^k, each number is divisible by 2 and always reaches
1.
a
element of (except 0) Natural Odd Numbers
n
and m elements of (except 0) All Natural Numbers
b
element of (except 0) Z {..-5,-3,-1,1,3,5,7...} +/- Odd numbers
k
elements of (except 0) N {1,2,3,4...} All Natural Numbers
an+b = 2^(k)
x,
m, k elements of N (excluding zero) {1,2,3,4...}
y
element of Z {..-5,-4,-3,-2,-1,0,1,2,3,4,5,7...}
x
and y < 2^(k-1)
a=2x+1,
n=m,
b=2y+1
a * n + b = 2^k
(2x+1)m
+ (2y +1) = 2^k
Examples:
if
k = 5 ==> y < (2^(k-1) -1)
==> y < 15
let y = 14
(2x+1)m + 2(14) + 1 = 2^5
(2x+1)m = 32 - 28 - 1
(2x+1)m = 3
factoring 3
We do not take 2x + 1 = 1 because x = 0
is not the element of N. Zero does not belong to the set where x is defined. So
we factor it into 2x + 1 = 3 and m = 1.
2x +1 = 3 and m = 1 and x = 1 ==> an+b ==> 3*1+29 ==> 32
==> Divided by 2, it reaches the number 1.
Thus, for the equation (2x+1)m + (2y +1) = 2^k when y <
(2^(k-1) -1), each value of the elements x, m, y and k Proves Collatz's Theorem is True.
****************************
Now
let's find out if the number 10.001 reaches 1.
If (2x+1)m = 10.001 then The power of 2
closest to 10.001 is 2^14 = 16384.
k=14 then y < (2^(k-1) -1) ==> y < (2^(13) -1)
(2x+1)m + 2y + 1 = 2^14
10.001 + 2y + 1 = 16384
2y = 16384 - 10.002, 2y= 6382, y = 3191
factoring 10001
2x+1 = 73, m = 137 ==> x = 36 ==> an+b ==> 73*137 + 2(3191) +1 ==>
16384
This proof is belongs to High Mathematician Oktay Haraççı and programmer by Gaye Haraççı.
Computer programs we made on this subject:
http://gayeer.brinkster.net/cassandra/collatz2/Father_Collatz2.aspx
http://gayeer.brinkster.net/cassandra/collatz/Father_Collatz.aspx
Keywords: Collatz, collatz conjecture solved, collatz conjecture, unsolved problem, collatz theorem, Wikipedia, Oktay Haraççı, Gaye Haraççı, Proof of the Collatz Conjecture, Lothar Collatz, Ulam, Kakutani, hailstone
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