## The Collatz Conjecture and Variants

Ever since I was introduced to the Collatz Conjecture in 1992 I have been attracted to the beauty of how such simplicity generates such structure. Well that and that it cycles too.

Yesterday on my lunch break at work I realized a variant of this famous dynamic system.

In the Collatz Conjecture or Hailstones as it is also known we look at the even-ness or odd-ness of a number. If odd we multiply by three and add one. If even we divide by two. The result of this process in irritation is that the value of 1 is always reached and a cycle of the values { 4 2 1 } are seen.

In my experiments with this fascinating dynamic system I have observed that there are cycles for all odd addend integers. With 3x+3 and a value of 1 the sequence is 1 > 6 > 3 > 12 , 6, 3 so the cycle then is { 12 6 3 } where the addend is 3

Now consider algebraic grouping of this for the addend of 1

Instead of (3x) + 1 let us do this 3(x+1).

Therefore we now see a cycle that has a sequence with the multiplier in it and not the addend in it.

Below is a copy of a post I made on mathstackexchange.com

I am seeking to know who may have reported this already so I don’t make a false claim by writing a proper paper introducing this form.

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As someone who has spent many man hours on this I thought to share an observation I made yesterday on my lunch break and to ask if it has been reported before. If not reported then an effort must be made.

We are familiar with the Collatz Conjecture aka Hailstones where we multiply by three then add one for when the state of the integer is odd. 3x+1 and x/2 for when the integer is even.

If we regroup this action and add one to an odd integer then multiply by three the cycle is attracted to the multiplier and not the addend. 3(x+1), x/2

{ 1 ,6 ,3 ,12 ,6 ,3 }

{ 2 ,1 ,6 ,3 ,12 ,6 ,3 }

{ 3 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 4 ,2 ,1 ,6 ,3 ,12 ,6 ,3 }

{ 5 ,18 ,9 ,30 ,15 ,48 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 6 ,3 ,12 ,6 ,3 }

{ 7 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 8 ,4 ,2 ,1 ,6 ,3 ,12 ,6 ,3 }

{ 9 ,30 ,15 ,48 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 10 ,5 ,18 ,9 ,30 ,15 ,48 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 11 ,36 ,18 ,9 ,30 ,15 ,48 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 12 ,6 ,3 ,12 ,6 ,3 }

{ 13 ,42 ,21 ,66 ,33 ,102 ,51 ,156 ,78 ,39 ,120 ,60 ,30 ,15 ,48 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 14 ,7 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 15 ,48 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 16 ,8 ,4 ,2 ,1 ,6 ,3 ,12 ,6 ,3 }

{ 17 ,54 ,27 ,84 ,42 ,21 ,66 ,33 ,102 ,51 ,156 ,78 ,39 ,120 ,60 ,30 ,15 ,48 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 18 ,9 ,30 ,15 ,48 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 19 ,60 ,30 ,15 ,48 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 20 ,10 ,5 ,18 ,9 ,30 ,15 ,48 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

So: If Odd do x+1 then multiply by three. If Even then divide by two.

Has anyone seen this before? I’ll write it up and put it online if not with a short complete C program attached.

I tested this with a 3321928 bit number. It has been tested with multiplier of two and three. The “attractor” then is the multiplier and not the addend as is seen in the Collatz form.

I will try 4,5,… soon.

mpz_set_ui( A, 3 ); // Set A to the multiplier you wish for A > 0

mpz_set_str( Y, “1\0”,10);

// Example C language code snippit using GMP bignum laibrary for(x=1;x<18446744073709551615u;x++) { mpz_set_ui(Z,x); w=0; printf("{ "); for(;;) { gmp_printf("%Zu ,",Z); if( mpz_odd_p(Z) ) { mpz_add_ui(Z,Z,1); mpz_mul(Z,Z,A); } else {mpz_divexact_ui(Z,Z,2);} if( mpz_cmp(Z,A) == 0 )w++; if(w==2){ gmp_printf("%Zu }",Z); printf("\n-----------\n");break;} } } I have enjoyed the Collatz conjecture since being introduced to it in 1992. I have worked out a great deal about it so I'm open to sharing what I know. In General I adore the cycle in mathematics. Always have. Thank You for your time. ------------------------------- So please let me know if there is prior art on this. My conjecture then is that for any multiplier there is a cycle with that multiplier in it. Again I am seeking prior art so as to not claim this as a discovery falsely. Ernst

Well, the Conjecture that “ALL” multipliers create a cycle where that multiplier will be in a cycle like we see with the Collatz or Hailstone problem was rather quickly dealt a blow. The Multiplier 9 for if odd 9(X+1) if even X/2 looks to be divergent. By that I mean it keeps growing in size and heads away from a cycle, from what I can tell. So while we have some cycles that look familiar to us such as the cycle for 3X+1 and 3X+3 when we look closely the cycles are being generated by a very different process.

So this is a second choice for us in studying the 3X+1 problem. It suggests we should review our rigid rules on order of precedence. 3X+1 and 3(X+1) give us two different universes to explore.

More important is a second example of that cycle structure proving again “the Universe doesn’t like One-zees.” I picked that saying up some place but it’s true.

I am working on a new type of sort for data compression right now but I am always interested in the realm of the Collatz Conjecture.

Ernst

Well, I experimented with the [A(x+1),X/2] form of the family of dynamic equations that Collatz Conjecture is.

I can say that the “parity structure” of the actions where we represent an odd action with a set bit {1} and the even action we represent with {0} in the stream of iterations for all numbers appears to cycle to {2,1} when A = 1; {4,2,1} for when A=2

I was able to use the [1(x-1),x/2] form to transform the Million Digit File and it was 1 to one and yet the two files were different. No compression on the new form but that is to be expected.

So it looks good in that we can study this form of the dynamic equation too along with the Collatz forms. Perhaps we will see clues in understanding Collatz by examining these “different acting” dynamic equations.

It is very very good to have a second validation of the cycle structure {4,2,1} for [3x+1,x/2] in [2(x+1,)x/2] Interesting that [3(x+1),x/2] reflects [3x+3,x/2] in cycle structure.

I think we have a clue here as to the mystery of the Collatz Conjecture. The [A(x+1),x/2] forms scale for x > 0 and seem to be divergent at A=9 so there is a possible example of a divergent system to check out.

By divergent I mean doesn’t cycle just keep on growing. Convergent is when it goes into a cycle. If I have these terms wrong I welcome correction.

So…. I am spending some time exploring this [A(x+y),x/2] form and it is like exploring the Collatz Conjecture all over.

I’m betting there is much more to realize.

Also I am very interested in any Prior Art on [A(x+1),x/2] since I would love to see what has been done with it if anything.

I put forth the suggestion that we may prove this Conjecture another way over on http://forum.objectivismonline.com/index.php?showtopic=21904&page=3

If we define the struggle between the 3x+1 and the x/2 as the x/2 winning and in general the value of x heads towards 1 then we can show that for any odd y in 3X+Y that a cycle of values exists and as such a reduction and the effect of the cycle of the odd Y we have a solution.

Do any of you work with this type of dynamic equation in ways other then trying to prove the path to one?

Well, my mind has been on vacation for a month. I guess sometimes it becomes necessary to take a break from things.

I am considering writing my second paper. I will ask the question of if the whole “Goes to One” concept is a red herring.

Interesting thing.

I have found a program I wrote years ago that found all the cycles in this dynamic system for all odd y values as in [3(x)+y,x/2]. I remember things a bit different so I must have edited the output to make the list I had for years of all the Cycles. I also have looked over the code again and am impressed with the work I had forgotten. I really thought things through on this one however, it may be time to make this effort multi-core and also to see if I can improve performance.

Here is a the output from the version dated 2005 and it reports a few of the Y values.

Attractor Finder program by Ernst Berg June 2005

Report for the Y of 1 is a power of 3 and a single attractor system

Report for the Y of 3 is a power of 3 and a single attractor system

The attractor >> 1 has been added. Run had 5 steps and the X is 1

The attractor >> 19 has been added. Run had 18 steps and the X is 3

The attractor >> 5 has been added. Run had 4 steps and the X is 5

The attractor >> 23 has been added. Run had 9 steps and the X is 23

The attractor >> 187 has been added. Run had 47 steps and the X is 123

The attractor >> 347 has been added. Run had 44 steps and the X is 171

^C

[Ernst@Unicorn6 Attractor_Finder]$ cd ~/Programming/Attractor_Finder/

[Ernst@Unicorn6 Attractor_Finder]$ ./attractor_finder

Attractor Finder program by Ernst Berg June 2005

Report for the Y of 1 is a power of 3 and a single attractor system

Report for the Y of 3 is a power of 3 and a single attractor system

The attractor >> 1 has been added. Run had 5 steps and the X is 1

The attractor >> 19 has been added. Run had 18 steps and the X is 3

The attractor >> 5 has been added. Run had 4 steps and the X is 5

The attractor >> 23 has been added. Run had 9 steps and the X is 23

The attractor >> 187 has been added. Run had 47 steps and the X is 123

The attractor >> 347 has been added. Run had 44 steps and the X is 171

Report for the Y of 5

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The value of Y is now 7

The attractor >> 5 has been added. Run had 9 steps and the X is 1

The attractor >> 7 has been added. Run had 4 steps and the X is 7

Report for the Y of 7

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The value of Y is now 9

Report for the Y of 9 is a power of 3 and a single attractor system

The attractor >> 1 has been added. Run had 9 steps and the X is 1

The attractor >> 13 has been added. Run had 28 steps and the X is 3

The attractor >> 11 has been added. Run had 4 steps and the X is 11

Report for the Y of 11

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The value of Y is now 13

The attractor >> 1 has been added. Run had 6 steps and the X is 1

The attractor >> 13 has been added. Run had 4 steps and the X is 13

The attractor >> 131 has been added. Run had 43 steps and the X is 19

The attractor >> 211 has been added. Run had 26 steps and the X is 99

The attractor >> 259 has been added. Run had 13 steps and the X is 123

The attractor >> 227 has been added. Run had 16 steps and the X is 147

The attractor >> 287 has been added. Run had 21 steps and the X is 159

The attractor >> 251 has been added. Run had 16 steps and the X is 163

The attractor >> 283 has been added. Run had 14 steps and the X is 283

The attractor >> 319 has been added. Run had 14 steps and the X is 319

Report for the Y of 13

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The value of Y is now 15

The attractor >> 57 has been added. Run had 20 steps and the X is 1

The attractor >> 3 has been added. Run had 5 steps and the X is 3

The attractor >> 15 has been added. Run had 6 steps and the X is 5

The attractor >> 69 has been added. Run had 11 steps and the X is 41

The attractor >> 561 has been added. Run had 49 steps and the X is 241

The attractor >> 1041 has been added. Run had 46 steps and the X is 337

Report for the Y of 15

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Well with Spring like summer already I have been spending a lot of time outdoors.

However, I think I could modernize the Cycle Finder program and share it GNU style.

When I wrote that attractor finder program I had two CPU’s but not two threads so now that Multi-Thread is common I could rewrite that application and share it with the Collatz community.

So perhaps by July I will have that done. There isn’t a list of cycles on-line any more since the host I had it on went down years ago.

I have an interesting development I am working on.

I will be devoting all my time to this for the foreseeable future.

This is just the same as changing the addend to 3, i.e., 3(x+1) = 3x + 3.

There may be a one to one with that. Honestly this has gone “cold” in the scope of my current interests.

Do you understand the parity language structure behind these dynamic equations? I can look into this again but I think I remember that this is not the same as 3x+3 parity language structure wise. I could be wrong so I will investigate and explain if you are interested.