New Collatz Conjecture Tool Box

[Rudy Toody]

I created a Mathematica tool box for anyone to use to try some extreme research on the 3x+1 conjecture. My functions go many magnitudes beyond what the current processes do.

Here is a link to my recent post on Mathematica Stack Exchange. I need many views from different IP addresses.

Thanks for viewing.

 

[Ken g6]

Looks interesting, but I can't get past the Mathematica expressions. Can you explain things more in English? Or standard mathematical notation? Or any lower-level programming language?

I gather than an "orbit" is the entire trajectory from n0 to (probably) 1. First, can you define a "sub-orbit"? I assume it's not a cycle.

I get where 3n+1 comes from, and 2n-1 (going "upstream"), but where does 2n+1 come from? I can see where (3n+1)/2 would appear, but why does (3/2)^n (1 + y) -1 appear? Is that somehow a recursion of (3n+1)/2? I don't see how that would work given that extra evens must be identified.

 

[Rudy Toody]

Looks interesting, but I can't get past the Mathematica expressions. Can you explain things more in English? Or standard mathematical notation? Or any lower-level programming language?

I gather than an "orbit" is the entire trajectory from n0 to (probably) 1. First, can you define a "sub-orbit"? I assume it's not a cycle.

I get where 3n+1 comes from, and 2n-1 (going "upstream"), but where does 2n+1 come from? I can see where (3n+1)/2 would appear, but why does (3/2)^n (1 + y) -1 appear? Is that somehow a recursion of (3n+1)/2? I don't see how that would work given that extra evens must be identified.

If you factor odd x+1 and look at the power of two, it defines the sub orbit. If it is 2^1, it is a descender and if it is 2^(something greater than 1), it is an ascender. Therefore, a descending sub orbit has one multiply. An ascending sub orbit has IntegerExponent[x+1,2] multiplies. The count of multiplications is the same as the count of directed edges in the graph data.

(3/2)^n (1 + y) -1 is the key to this entire process!

I was curious about the structure of sequences of ascending numbers, so I nested ((3x+1)/2) eight deep to get (3^8-2^8+3^8 x)/2^7. Then I substituted y for the exponents and simplified to get 2((3/2)^y (1 + x) -1) (I dropped the leading 2 because it gets cancelled during calculations). When I set x to a number I knew started a sub orbit of 8, and set y to 8, I got an even number which when the powers of 2 were removed, equals the last number of the sub orbit, as expected. So that provided the omegaSubOrbit[x] function. As I said above, the (x+1) from that simplification is the critical observation.

IntegerExponent[x+1,2] is a [I]Mathematica[/I] function that counts the trailing zeroes
in base two, which give us the exponent of two we use to reduce to an odd number

2n+1 is creating odd numbers. For the functions with limits, (i.e., 1000000) means 1000000 odd numbers.

The key is to remember we use only odd numbers for x.

 

[Ken g6]

Interesting, so you found a way to run through an ascender series (sub-orbit) in one step, in a similar way to the fairly obvious descender series step.

If you factor odd x+1 and look at the power of two, it defines the sub orbit.

How did you determine this?

I suspect this may be a special case of the array precalculation optimization. I also suspect that it's not as fast - though it's less memory-intensive. I tried alternating ascenders/descenders in OpenOffice, and the length for 47 was only about half as long. Precalculating with, e.g. 16 bits reduces the length of the calculation by that factor (16). (Though 2^16 precalculations are necessary.)

 

[Rudy Toody]

If you factor odd x+1 and look at the power of two, it defines the sub orbit.

This was found at the same time as the omegOrbit calculation was found. I took all 8 numbers of the ascending sequence and applied them against the simplified 8-deep formula and found that the denominators were increasing powers of two, i.e., 1, 2, 4, 8 ... and that the value I needed was the power of two that got cancelled in both the numerator and denominator. Since (x+1) was the only even number in the numerator of the simplified 8-deep formula, I factored each (x+1) for each of the eight sample numbers and found that the IntegerExponent[x+1,2] function was just what I needed. Everything I found was a latent pattern extracted from the (3x+1)/2 sequences.

 

[Philippart]

I wanted to upvote your post, but I noticed that they don't transfer the reputation from their sister websites... I will do if I manage to transfer my score from "crossvalidated", the statistics community at stackexchange

 

[Rudy Toody]

I wanted to upvote your post, but I noticed that they don't transfer the reputation from their sister websites... I will do if I manage to transfer my score from "crossvalidated", the statistics community at stackexchange

Thanks, Tom:thumbsup: Did you check my profile? I'll have to check "crossvalidated" to see how they are treating you

 

[Philippart]

Thanks, Tom:thumbsup: Did you check my profile? I'll have to check "crossvalidated" to see how they are treating you

haha I am "Tomanband" over there I have experience in Matlab and R, I guess Mathematica code shouldn't be too hard to port to these languages.

 

[Rudy Toody]

haha I am "Tomanband" over there I have experience in Matlab and R, I guess Mathematica code shouldn't be too hard to port to these languages.

I haven't used either one, but there have been a few references to them. I'll check out Tomanband and give you a vote or two.DONE! My statement about the patterns and the math also applies to patterns and statistics.

 

[Rudy Toody]

I guess I'm going to have to code in C. Mathematica doesn't compile for Integers Of Unusual Size.

 

[Rudy Toody]

On this link provided by Ken_g6, is a section for a partial proof of the Syracuse Conjecture, Since my functions only deal with odd numbers they describe this conjecture.

In examining the partial proof, I have found a conflict in bullet three and the problematic statement.

• This is equivalent to my definition of a descender.
• This is equivalent to what I call the lonely uniques--numbers which are never targets of descenders.
• This is equivalent to the ascenders which always have p>1. And ascenders are always 3=x mod 4.

The problem with the problematic statement is the 1=x mod 4, because the problem is, if that is what is expected, it won't happen because it has already been shown by bullet one.

 

[Rudy Toody]

I suspect this may be a special case of the array precalculation optimization. I also suspect that it's not as fast - though it's less memory-intensive.

I compared my un-optimized functions against a WU from Collatz (which takes 4hr) for over 12hr and wasn't even half-way through when I cancelled. I don't know what improvements using C and GMP can provide, but it would have to be greater than 12-fold.

So, I'm going to use the functions for their original purpose---to explain different patterns when I graph the sub orbits. The most interesting of those patterns is when we graph (say) 10,000 odd numbers (skipping all non-unique sub orbits) we get 13,333 directed edges. I'm hoping to explain why it is a proportion of 1/3. This is mentioned in a few papers as an open question.

And, as I mentioned in an earlier post, the incomplete proof of the Syracuse Conjecture could be improved using these patterns. Though, I don't know if that would lead to an actual proof.