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November 23rd, 2016, 01:24 AM   #1
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Collatz Conjecture bounds

Cycle bounds

Collatz Conjecture

I have an update about what I was trying to find for so long.

(I am still kinda stuck with finishing my proof on a few things.)

I think I said when I will find a limit to x by y I will share it

x = number of even values in a cycle
y = number of odd values in a cycle

x<8/9*floor(1.5^y)-y

I know it's really high but it works and is helpful on a few cases.


Last edited by skipjack; November 23rd, 2016 at 07:34 PM.
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May 16th, 2017, 01:00 AM   #2
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anyone with lower bounds??? i think this is the best results

positive cycles:

x = roundup ( y * ln3/ln2 )

negative cycles:

x = rounddown ( y * ln3/ln2 )

(-1,-2)
(1,2,4)
(-5,-14,-7,-20,-10)
(−17,−50,−25,−74,−37,−110,−55,−16 4 ,−82,−41,−122,−61,−182,−91,−272,−1 36,−68,−34)
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June 30th, 2017, 09:54 PM   #3
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i can't believe it after 2 years it was right under my nose

d=2^x-3^y

if d=1 then you have 3 solutions

(-1,-2)
(1,2,4)
(-5,-14,-7,-20,-10)

if d>1 then we know we have atlease 1 solution

(−17,−50,−25,−74,−37,−110,−55,−16 4 ,−82,−41,−122,−61,−182,−91,−272,−1 36,−68,−34)

if this is the only solution for d>1 then the collatz conjecture is true

now the thing is i didnt know why this is for x=11 and y=7

and why this should be the only solution

-----------------------

the minimum value for any cycle is

12k+7
12k+11

or

96k+7
96k+31
96k+79
96k+91
96k+47
96k+59
96k+71
96k+95

etc ...

---------------------------

because of

minimum (mod 12) = 7
minimum (mod 12) = 11

this force it to be d=2^11-3^7

i love Number Theory

because of simple things like this

-----------------

this not proving the collatz conjecture but this explain why the extra solution must be x=11 and y=7

-----------------

its just like if d=1 then you have 3 solutions because |2^x-3^y|=1 gives

x=1,y=1
x=2,y=1
x=3,y=2

Last edited by isaac; June 30th, 2017 at 10:10 PM.
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July 13th, 2017, 01:29 AM   #4
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new update

let y be the total odd values in a cycle
let max be the cycle maximum value for a cycle with y odd values

for y=5 max=13y*1.5^y
for y=306 max=12.8y*1.5^y
for y>=2966 max=1.6y*1.5^y

nice ah?

i proved that today

Last edited by isaac; July 13th, 2017 at 01:38 AM.
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