I’ve just uploaded to the arXiv my paper “Almost all Collatz orbits attain almost bounded values“, submitted to the proceedings of the Forum of Mathematics, Pi. In this paper I returned to the topic of the notorious Collatz conjecture (also known as the conjecture), which I previously discussed in this blog post. This conjecture can be phrased as follows. Let denote the positive integers (with the natural numbers), and let be the map defined by setting equal to when is odd and when is even. Let be the minimal element of the Collatz orbit . Then we have

Conjecture 1 (Collatz conjecture)One has for all .

Establishing the conjecture for all remains out of reach of current techniques (for instance, as discussed in the previous blog post, it is basically at least as difficult as Baker’s theorem, all known proofs of which are quite difficult). However, the situation is more promising if one is willing to settle for results which only hold for “most” in some sense. For instance, it is a result of Krasikov and Lagarias that

for all sufficiently large . In another direction, it was shown by Terras that for almost all (in the sense of natural density), one has . This was then improved by Allouche to for almost all and any fixed , and extended later by Korec to cover all . In this paper we obtain the following further improvement (at the cost of weakening natural density to logarithmic density):

Theorem 2Let be any function with . Then we have for almost all (in the sense of logarithmic density).

Thus for instance one has for almost all (in the sense of logarithmic density).

The difficulty here is one usually only expects to establish “local-in-time” results that control the evolution for times that only get as large as a small multiple of ; the aforementioned results of Terras, Allouche, and Korec, for instance, are of this type. However, to get all the way down to one needs something more like an “(almost) global-in-time” result, where the evolution remains under control for so long that the orbit has nearly reached the bounded state .

However, as observed by Bourgain in the context of nonlinear Schrödinger equations, one can iterate “almost sure local wellposedness” type results (which give local control for almost all initial data from a given distribution) into “almost sure (almost) global wellposedness” type results if one is fortunate enough to draw one’s data from an *invariant measure* for the dynamics. To illustrate the idea, let us take Korec’s aforementioned result that if one picks at random an integer from a large interval , then in most cases, the orbit of will eventually move into the interval . Similarly, if one picks an integer at random from , then in most cases, the orbit of will eventually move into . It is then tempting to concatenate the two statements and conclude that for most in , the orbit will eventually move . Unfortunately, this argument does not quite work, because by the time the orbit from a randomly drawn reaches , the distribution of the final value is unlikely to be close to being uniformly distributed on , and in particular could potentially concentrate almost entirely in the exceptional set of that do not make it into . The point here is the uniform measure on is not transported by Collatz dynamics to anything resembling the uniform measure on .

So, one now needs to locate a measure which has better invariance properties under the Collatz dynamics. It turns out to be technically convenient to work with a standard acceleration of the Collatz map known as the *Syracuse map* , defined on the odd numbers by setting , where is the largest power of that divides . (The advantage of using the Syracuse map over the Collatz map is that it performs precisely one multiplication of at each iteration step, which makes the map better behaved when performing “-adic” analysis.)

When viewed -adically, we soon see that iterations of the Syracuse map become somewhat irregular. Most obviously, is never divisible by . A little less obviously, is twice as likely to equal mod as it is to equal mod . This is because for a randomly chosen odd , the number of times that divides can be seen to have a geometric distribution of mean – it equals any given value with probability . Such a geometric random variable is twice as likely to be odd as to be even, which is what gives the above irregularity. There are similar irregularities modulo higher powers of . For instance, one can compute that for large random odd , will take the residue classes with probabilities

respectively. More generally, for any , will be distributed according to the law of a random variable on that we call a *Syracuse random variable*, and can be described explicitly as

where are iid copies of a geometric random variable of mean .

In view of this, any proposed “invariant” (or approximately invariant) measure (or family of measures) for the Syracuse dynamics should take this -adic irregularity of distribution into account. It turns out that one can use the Syracuse random variables to construct such a measure, but only if these random variables stabilise in the limit in a certain total variation sense. More precisely, in the paper we establish the estimate

for any and any . This type of stabilisation is plausible from entropy heuristics – the tuple of geometric random variables that generates has Shannon entropy , which is significantly larger than the total entropy of the uniform distribution on , so we expect a lot of “mixing” and “collision” to occur when converting the tuple to ; these heuristics can be supported by numerics (which I was able to work out up to about before running into memory and CPU issues), but it turns out to be surprisingly delicate to make this precise.

A first hint of how to proceed comes from the elementary number theory observation (easily proven by induction) that the rational numbers

are all distinct as vary over tuples in . Unfortunately, the process of reducing mod creates a lot of collisions (as must happen from the pigeonhole principle); however, by a simple “Lefschetz principle” type argument one can at least show that the reductions

are mostly distinct for “typical” (as drawn using the geometric distribution) as long as is a bit smaller than (basically because the rational number appearing in (3) then typically takes a form like with an integer between and ). This analysis of the component (3) of (1) is already enough to get quite a bit of spreading on (roughly speaking, when the argument is optimised, it shows that this random variable cannot concentrate in any subset of of density less than for some large absolute constant ). To get from this to a stabilisation property (2) we have to exploit the mixing effects of the remaining portion of (1) that does not come from (3). After some standard Fourier-analytic manipulations, matters then boil down to obtaining non-trivial decay of the characteristic function of , and more precisely in showing that

for any and any that is not divisible by .

If the random variable (1) was the sum of independent terms, one could express this characteristic function as something like a Riesz product, which would be straightforward to estimate well. Unfortunately, the terms in (1) are loosely coupled together, and so the characteristic factor does not immediately factor into a Riesz product. However, if one groups adjacent terms in (1) together, one can rewrite it (assuming is even for sake of discussion) as

where . The point here is that after conditioning on the to be fixed, the random variables remain independent (though the distribution of each depends on the value that we conditioned to), and so the above expression is a *conditional* sum of independent random variables. This lets one express the characeteristic function of (1) as an *averaged* Riesz product. One can use this to establish the bound (4) as long as one can show that the expression

is not close to an integer for a moderately large number (, to be precise) of indices . (Actually, for technical reasons we have to also restrict to those for which , but let us ignore this detail here.) To put it another way, if we let denote the set of pairs for which

we have to show that (with overwhelming probability) the random walk

(which we view as a two-dimensional renewal process) contains at least a few points lying outside of .

A little bit of elementary number theory and combinatorics allows one to describe the set as the union of “triangles” with a certain non-zero separation between them. If the triangles were all fairly small, then one expects the renewal process to visit at least one point outside of after passing through any given such triangle, and it then becomes relatively easy to then show that the renewal process usually has the required number of points outside of . The most difficult case is when the renewal process passes through a particularly large triangle in . However, it turns out that large triangles enjoy particularly good separation properties, and in particular afer passing through a large triangle one is likely to only encounter nothing but small triangles for a while. After making these heuristics more precise, one is finally able to get enough points on the renewal process outside of that one can finish the proof of (4), and thus Theorem 2.

## 450 comments

Comments feed for this article

31 August, 2021 at 9:16 am

Wulf RehderCalculating the Total Stopping Time N(T):

[Note: In my last reply of August 29, above, the final sentence has a typo: “ratio (m+k)^N” should read “ratio (m+k)/N”.]

My previous comments on the Collatz Conjecture in this thread have shown that the stopping time N=N(S) can be determined by the proportion of divisions by 2 and the constant t, which is related to the Cantor fractal dimension r=log2/log3 or, equivalently, to the Sierpinski fractal dimension s=1/r, by the relationship t=1/(r+1)=s/(s+1)=0.613147…. To wit: If m is the number of multiplications by 3 and m+k the number of divisions by 2 up to the index N=m+m+k, then N=N(S) is the stopping time of this sequence iff at this N the proportion

(*) (m+k)/N >= t for the first time.

The starting element c(0) (the “seed”) of the Collatz sequence does not explicitly appear in this formula. This characterization of the stopping time in terms of a fractal dimension has been verified with all available data, for instance https://en.wikipedia.org/wiki/Collatz_conjecture#Empirical_data as well as for the Collatz-relevant sequences A006884 and A006577 in the OEIS.

In a similar way, the total stopping time N(T), the index at which the Collatz sequence hits down to 1, can be characterized in terms of s, r, and t. Here, however, the seed c(0) figures explicitly. Again we are looking for the excess k of the number of divisions by 2 over the number m of multiplications by 3, such that this excess pushes the seed c(0) down to 1:

(**) c(0) 3^m/2^(m+k) N(S) and

(***) [k+m-log(c(0))/log2]/[N-log(c(0))/log2] >= t,

with t being the threshold defined above.

Comparing (***) to (*) shows how the seed c(0) enters the calculation of the total stopping time explicitly. [N(T)>N(S) is added because for some small N=1,2,3, … = C, or (k+m)-sm >= D,

with C>0 and D>0 depending on the seed c(0): The Collatz sequence with seed c(0) reaches 1 after m+m+k steps, iff the relationship between the m+k divisions by 2 and the m multiplications by 3 are given by the fractal dimensions r and s in (****).

None of this proves the Collatz Conjecture, of course. However, the somewhat elusive conjecture “The sequence eventually reaches 1” can be more concretely expressed in terms of the inequalities (***) or (****). A counter-example to the Collatz Conjecture would require a seed c(0) such that, for instance, r(k+m)-m < C for all N=m+m+k..

P.S.: A reader who intends to “Rate Down” this or any other post – which seems to happen almost automatically in this thread, quite drastically even for contributions by Dr. Tao – is politely requested to give a reason in a reply, or better even, a counter-example. I’d be happy to explain further. Thanks!

31 August, 2021 at 11:25 pm

Wulf RehderCalculating the Total Stopping Time N(T)

For mysterious reasons a part of the text was excised in the copy-paste process of the above posting: Instead of the text from “Comparing (***) to (*) …” and “… r and s in (****)” the correct text should read:

“Comparing (***) to (*) shows how the seed c(0) enters the calculation of the total stopping time explicitly. [N(T)>N(S) is added because for some small N=1,2,3, … = C, or (k+m)-sm >= D,

with C>0 and D>0 depending on the seed c(0): The Collatz sequence with seed c(0) reaches 1 after m+m+k steps, iff the relationship between the m+k divisions by 2 and the m multiplications by 3 are given by the fractal dimensions r and s in (****).”

31 August, 2021 at 11:43 pm

Wulf RehderCalculating the Total Stopping Time N(T)

I found the mysterious reason: There was a “smaller than” symbol in the text, which is fatal in certain versions of wordpress. Now then: Instead of the text from “Comparing (***) to (*) …” and “… r and s in (****)” the correct text should read:

“Comparing (***) to (*) shows how the seed c(0) enters the calculation of the total stopping time explicitly. [N(T)>N(S) is added because for some small N=1,2,3, … smaller than N(S) the ratio (***) may be large or even negative.] This too has been corroborated with a wide variety of interesting seeds c(0)=27, 97, 871, 6 171, 77 031, 837 799, … and OEIS sequences.

Defining the constants A=log(c(0))/log2, B=tA, (A-B)/t=C, D=C/r makes (***) more compact:

(****) r(k+m)-m >= C, or (k+m)-sm >= D,

with C>0 and D>0 depending on the seed c(0): The Collatz sequence with seed c(0) reaches 1 after m+m+k steps, iff the relationship between the m+k divisions by 2 and the m multiplications by 3 are given by the fractal dimensions r and s in (****).”

Sorry about this.

20 September, 2021 at 7:40 am

De WiWulf Rehder, I am writing some code on Collatz Sequences on Tool Control Language, TCL. Thanks for your previous (TAO) posts on subject. Is there a “formula” or psuedocode steps to estimate the number of elements or steps of a Collatz Sequence before computing the Collatz Sequence by the computer. The complement to your Rehder_limt is 0.3868528072345415 or 38.6 percent. Might useful in some situations. Thank you.

Tool Control Language = TCL.

—————————————————-

proc collatz_sequences_Krasikov_Lagarias_limit {limit} {

if { $limit <= 1 } {

return -code error "The limit must be larger than 1"

}

expr {$limit** 0.84}

}

# Krasikov and Lagarias limit 4 3.204

# Krasikov and Lagarias limit 5 3.864

# Krasikov and Lagarias limit 10 6.918

# Krasikov and Lagarias limit 20 12.384

# Krasikov and Lagarias limit 50 26.738

set Rehder_limit_collatz_sequences [ return [ expr { (log(3)/log(2))/(log(3)/log(2)+1.)} ] ]

# returns 0.61314719276545848 or or 61.3 percent

set Rehder_limit_complement [ return [ expr { 1.- (log(3)/log(2))/(log(3)/log(2)+1.)} ] ]

# returns 0.3868528072345415 or 38.6 percent

20 September, 2021 at 8:04 am

michaelmrossThere is the special case of 4n+1 being always a sequence with one more step than sequence n – really the same sequence with one earlier step. Now also remember, that the stopping time of every number in a sequence is known from the first number in the sequence. Every number can only be a member of one sequence.

Besides that there are nonrecursive functions – that is, ones that eliminate division or exponentiation, so they have linear running time, such as:

One may also consider the possibility that sequences are actually superfluous, and the conjecture concerns the finitude of the input and output of a table with three congruences and three operations. https://youtu.be/q1F50FTpg8A

22 September, 2021 at 7:14 am

De WiSubject: Possible Uses for Collatz Sequences Conjecture?

#### A novice programmer (me) has received offsite query from “senior programmer in charge” ..>>> .. ” While I am fascinated by the Collatz problem, I do not see much practical use for it. In what kind of computer applications would you use this?”

—————————

#### Answer, As understood here, the Collatz Sequences Conjecture is called pure mathematics. Paul Erdős said about the Collatz conjecture: “Mathematics may not be ready for such problems.” The Collatz Sequences Conjecture is at the forefront of human mathematics. From what an outsider can tell, the Collatz Sequences are being generated with supercomputers and distributed computing algoritms on the numeric issues. Apparently, the Collatz Sequences are as valid a test for computing machines, parallel algoritms, and distributed computing algoritms as the 1) search for pi , 2) search for highest prime number, 3) Twin Prime Conjecture, and 4) factoring numbers . The novice has written TCL hacks on all five subjects. In 2020, starting values up to 2**268 approximating 2.95*1020 have been checked in the Collatz conjecture. I suppose the question is valid, if one is advocating spending time, salaried employees, and resources on pure mathematics and the Collatz Sequences Conjecture to decision making nodes. .I am a retired engineer. Maybe the pure mathematicians among us may offer some more practical uses for the Collatz Sequences Conjecture?

Samples from Tool Control Language = TCL.

—————————————————-

proc collatz_sequences_Krasikov_Lagarias_limit {limit} {

if { $limit <= 1 } {

return -code error "The limit must be larger than 1"

}

expr {$limit** 0.84}

}

# Krasikov and Lagarias limit 4 3.204

# Krasikov and Lagarias limit 5 3.864

# Krasikov and Lagarias limit 10 6.918

# Krasikov and Lagarias limit 20 12.384

# Krasikov and Lagarias limit 50 26.738

set Rehder_limit_collatz_sequences [ return [ expr { (log(3)/log(2))/(log(3)/log(2)+1.)} ] ]

# returns 0.61314719276545848 or or 61.3 percent

set Rehder_limit_complement [ return [ expr { 1.- (log(3)/log(2))/(log(3)/log(2)+1.)} ] ]

# returns 0.3868528072345415 or 38.6 percent

19 October, 2021 at 3:50 pm

E FDear All,

Today I posted a document on Reddit regarding the Collatz conjecture (Family Tree Problem). I have tried to approach the Collatz conjecture in a different way with the idea of discovering new patterns. I have a strongly believe that the mathematical functions f(n) = 3n – 1, f(n) = 2n – 1, f(n) = 3n + 1 and f(n) = 4n + 1 are closely related to the Family Tree Anomalies (Loops related to f(n) = 3n + 1) especially for n = 0, n = -2, n = -6 and n = -8. Please take a look at it if you have any spare time these days.

I tried to copy some details of the document for you, but maybe the readability is not best.

Family Tree “Genealogy” Problem (Reverse Collatz Conjecture)

By EF

October 15, 2021

*THE MATHEMATICAL FUNCTIONS RELATED TO THE FAMILY TREE PROBLEM ARE:

f(n) = 3n + 1; to generate Parent of n

f(n) = n + [(n – 1) / 3]; to generate Child (firstborn) of n

f(n) = n – [(n + 1) / 3]; to generate Child (firstborn) of n

f(n) = 2n; to generate Child (firstborn) of n (n divisible by 3 doesn’t have any odd children)

f(n) = 4n + 1; to generate Sibling of n

f(m,n) = {[(4^(m-1)).(3n + 1)] – 1} / 3; to generate Birth order of e.g. Siblings or Children of n, where m = Birth order

*EXAMPLE PERSON N = 11

9 Grandchild ; f(9) = 2n = 18 (n divisible by 3 doesn’t have any odd

children)

x3 + 1 =

28 f(9) = 3n + 1

/2 =

14 f(9) = (3n + 1) / 2

/2 =

7 Child (firstborn) ; f(9) = ((3n + 1) / 2) / 2 ; f(11) = n – [(n + 1) / 3] = 7

(Firstborn of 11)

x3 + 1 =

22 f(7) = 3n + 1

/2 =

11 Person ; f(7) = (3n + 1) / 2 ; f(17) = n – [(n + 1) / 3] = 11 (Firstborn of 17)

x3 + 1 =

34 f(11) = 3n + 1

/2 =

17 Parent ; f(11) = (3n + 1) / 2 ; f(13) = n + [(n – 1) / 3] = 17 (Firstborn of 13)

x3 + 1 =

52 f(17) = 3n + 1

/2 =

26 f(17) = (3n + 1) / 2

/2 =

13 Grandparent ; f(17) = ((3n + 1) / 2) / 2 ;

f(2,5) = {[(4^(m-1)).(3n + 1)] – 1} / 3 = 13 (Second Child of 5)

x3 + 1 =

40 f(13) = 3n + 1

/2 =

20 f(13) = (3n + 1) / 2

/2 =

10 f(13) = ((3n + 1) / 2) / 2

/2 =

5 Great-Grandparent ; f(13) = (((3n + 1) / 2) / 2) / 2 ;

f(2,1) = {[(4^(m-1)).(3n + 1)] – 1} / 3 = 5 (Second Child of 1)

x3 + 1 =

16 f(5) = 3n + 1

/2 =

8 f(5) = (3n + 1) / 2

/2 =

4 f(5) = ((3n + 1) / 2) / 2

/2 =

2 f(5) = (((3n + 1) / 2) / 2) / 2

/2 =

1 Great-Great-Grandparent and Sibling of Great-Grandparent are the

same person (Parent and Firstborn Child are the same person)

f(5) = ((((3n + 1) / 2) / 2) / 2) / 2

x3 + 1 =

4 f(1) = 3n + 1

/2 =

2 f(1) = (3n + 1) / 2

/2 =

1 Progenitor (Founder of the Collatz family); 1 is Parent of Firstborn Child

1 and Grandparent of Firstborn Grandchild 1 (Parent and Firstborn

Child and Firstborn Grandchild are the same person namely 1)

f(1) = ((3n + 1) / 2) / 2

*EXAMPLE PERSON N = 11

f(m,n) = {[(4^(m-1)).(3n + 1)] – 1} / 3 ; Birth order of siblings or children of n ; where m = Birth Order

Siblings of 9 (Grandsons and Granddaughters of 11)

37 (second sibling); 149 (third sibling); 597 (fourth sibling); 2389 (fifth sibling) f(9) = 4n + 1

(m,n) = (1,9) = 9; (m,n) = (2,9) = 37; (m,n) = (3,9) = 149; (m,n) = (4,9) = 597; (m,n) = (5,9) = 2389

Siblings of 7 (Sons and Daughters of 11)

29 (second sibling); 117 (third sibling); 469 (fourth sibling); 1877 (fifth sibling) f(7) = 4n + 1

(m,n) = (1,7) = 7; (m,n) = (2,7) = 29; (m,n) = (3,7) = 117; (m,n) = (4,7) = 469; (m,n) = (5,7) = 1877

Siblings of 11 (Brothers and Sisters of 11)

45 (second sibling); 181 (third sibling); 725 (fourth sibling); 2901 (fifth sibling) f(11) = 4n + 1

(m,n) = (1,11)= 11; (m,n) = (2,11) = 45; (m,n) = (3,11) = 181; (m,n) = (4,11) = 725; (m,n) = (5,11) = 2901

Siblings of 17 (Aunts and Uncles of 11)

69 (second sibling); 277 (third sibling); 1109 (fourth sibling); 4437 (fifth sibling) f(17) = 4n + 1

(m,n) = (1,17) = 17; (m,n) = (2,17) = 69; (m,n) = (3,17) = 277; (m,n) = (4,17) = 1109; (m,n) = (5,17) = 4437

Siblings of 13 (Grandaunts and Granduncles of 11)

3 (first sibling); 53 (third sibling); 213 (fourth sibling); 853 (fifth sibling) f(13) = 4n + 1

(m,n) = (1,13) = 13; (m,n) = (2,13) = 53; (m,n) = (3,13) = 213; (m,n) = (4,13) = 853; (m,n) = (5,13) = 3413

(m,n) = (1,3) = 3; (m,n) = (2,3) = 13; (m,n) = (3,3) = 53; (m,n) = (4,3) = 213; (m,n) = (5,3) = 853

Siblings of 5 (Great-Grandaunts and Great-Granduncles of 11)

1 (first sibling); 21 (third sibling); 85 (fourth sibling); 341 (fifth sibling) f(5) = 4n + 1

(m,n) = (1,5) = 5; (m,n) = (2,5) = 21; (m,n) = (3,5) = 85; (m,n) = (4,5) = 341; (m,n) = (5,5) = 1365

(m,n) = (1,1) = 1; (m,n) = (2,1) = 5; (m,n) = (3,1) = 21; (m,n) = (4,1) = 85; (m,n) = (5,1) = 341

Siblings of 1

5 (second sibling); 21 (third sibling); 85 (fourth sibling); 341 (fifth sibling)

(m,n) = (1,1) = 1; (m,n) = (2,1) = 5; (m,n) = (3,1) = 21; (m,n) = (4,1) = 85; (m,n) = (5,1) = 341

The loops regarding f(n) = 3n + 1 can be explained as anomalies related to the Family Tree.

*LOOPS OF F(N) = 3N + 1 (ANOMALIES IN FAMILY TREES)

f(n) = 3n + 1 f(n) = n + [(n – 1) / 3] f(n) = n – [(n + 1) / 3] f(n) = 4n + 1

Parent of n Child (firstborn) of n Child (firstborn) of n Siblings of n

n = 1

4 1 5

/ 2 21

2 85

/ 2 341

1

1 is Parent of Firstborn Child 1 and Grandparent of Firstborn Grandchild 1 (Parent and Firstborn Child and Firstborn Grandchild are the same person namely 1)

n = -1

-2 -1 -3

/ 2 -11

-1 -43

-1 is Parent of Firstborn Child -1 and Grandparent of Firstborn Grandchild -1

(Parent and Firstborn Child and Firstborn Grandchild are the same person namely -1)

n = -5

-14 -7 -19

/ 2 -75

-7 -299

-7 is Parent of Firstborn Child -5 and Grandparent of Firstborn Grandchild -7 (Grandparent and Firstborn Grandchild are the same person namely -7)

n = -7

-20 -5 -27

/ 2 -107

-10 -427

/ 2

-5

-5 is Parent of Firstborn Child -7 and Grandparent of Firstborn Grandchild -5 (Grandparent and Firstborn Grandchild are the same person namely -5)

n = -23

-68 -31 (-23 is Firstborn and -91 is Second Child of -17) /2

-34 -91

/ 2 -363

-17 -1451

n = -17

-50 -23 (-17 is Parent of -23) -67

/ 2 -267

-25 -1067

n = -25

-74 (-25 is Parent of -17) -17 -99

/ 2 -395

-37 -1579

n = -37

-110 (-37 is Grandparent of -17) -25 -147

/ 2 -587

-55 -2347

n = -55

-164 -37 (-55 is Great-Grandparent of -17) -219

/ 2 -875

-82 -3499

/2

-41

n = -41

-122 -55 (-41 is Great-Great-Grandparent of -17)

/ 2 -163

-61 -651

-2603

n = -61

-182

/ 2

-91

-61 is Great-Great-Great-Grandparent of -17 and Great-Great-Great-Great-Grandparent of -91

-91 is Great-Great-Great-Great-Grandparent of -17 and Great-Great-Great-Great-Great-Grandparent of -91

-91 is Second Child of -17 and Great-Great-Great-Great-Grandchild of -61

(‘Great-Great-Great-Great-Great-Grandchild’ and ‘Great-Great-Great-Great-Great-Grandparent’ are the same person namely -91)

n = -91

-272

/ 2

-136

/ 2

-68

/ 2

-34

/ 2

-17

*THE FUNCTIONS F(N) = 3N – 1, F(N) = 2N – 1, F(N) = 3N + 1 AND F(N) = 4N + 1 ARE CLOSELY RELATED TO THE FAMILY TREE ANOMALIES (LOOPS RELATED TO F(N) = 3N + 1) ESPECIALLY FOR N = 0, N = -2, N = -6 AND N = -8.

The difference of P and C is n The difference of C and P is n

(3n -1) – (2n -1) = n (4n + 1) – (3n + 1) = n

Parent Child Parent Child

f(n) = 3n – 1 f(n) = 2n – 1 n f(n) = 3n + 1 f(n) = 4n + 1

-106 -71 -35 -104 -139

-103 -69 -34 -101 -135

-100 -67 -33 -98 -131

-97 -65 -32 -95 -127

-94 -63 -31 -92 -123

-91 -61 -30 -89 -119

-88 -59 -29 -86 -115

-85 -57 -28 -83 -111

-82 -55 -27 -80 -107

-79 -53 -26 -77 -103

-76 -51 -25 -74 -99

-73 -49 -24 -71 -95

-70 -47 -23 -68 -91

-67 -45 -22 -65 -87

-64 -43 -21 -62 -83

-61 -41 -20 -59 -79

-58 -39 -19 -56 -75

-55 -37 -18 -53 -71

-52 -35 -17 -50 -67

-49 -33 -16 -47 -63

-46 -31 -15 -44 -59

-43 -29 -14 -41 -55

-40 -27 -13 -38 -51

-37 -25 -12 -35 -47

-34 -23 -11 -32 -43

-31 -21 -10 -29 -39

-28 -19 -9 -26 -35

-25 -17 -8 -23 -31

-22 -15 -7 -20 -27

-19 -13 -6 -17 -23

-16 -11 -5 -14 -19

-13 -9 -4 -11 -15

-10 -7 -3 -8 -11

-7 -5 -2 -5 -7

-4 -3 -1 -2 -3

-1 -1 0 1 1

2 1 1 4 5

5 3 2 7 9

8 5 3 10 13

11 7 4 13 17

14 9 5 16 21

17 11 6 19 25

20 13 7 22 29

23 15 8 25 33

26 17 9 28 37

29 19 10 31 41

32 21 11 34 45

35 23 12 37 49

38 25 13 40 53

41 27 14 43 57

44 29 15 46 61

47 31 16 49 65

50 33 17 52 69

53 35 18 55 73

56 37 19 58 77

59 39 20 61 81

62 41 21 64 85

65 43 22 67 89

68 45 23 70 93

71 47 24 73 97

74 49 25 76 101

Intersection Point f(n) = 3n + 1 and f(n) = 4n + 1 (0 ; 1)

Intersection Point f(n) = 2n – 1 and f(n) = 3n + 1 (-2 ; -5)

Intersection Point f(n) = 2n – 1 and f(n) = 3n – 1 (0 ; -1)

Intersection Point f(n) = 3n – 1 and f(n) = 2n (1 ; 2)

Intersection Point f(n) = 3n + 1 and f(n) = 2n (-1 ; -2)

Intersection Point f(n) = 2n – 1 and f(n) = n + [(n – 1) / 3] (1 ; 1)

Intersection Point f(n) = 2n – 1 and f(n) = 4n + 1 (-1 ; -3)

Intersection Point f(n) = 2n – 1 and f(n) = n – [(n + 1) / 3] (0,5 ; 0)

Intersection Point f(n) = 4n + 1 and f(n) = 2n and f(n) = n + [(n – 1) / 3]

(-0,5 ; -1)

Intersection Point f(n) = n + [(n – 1) / 3] and f(n) = n – [(n + 1) / 3] (0 ; -1/3)

Intersection Point f(n) = 3n – 1 and f(n) = 4n + 1 (-2 ; -7)

Intersection Point f(n) = 3n – 1 and f(n) = n + [(n – 1) / 3] (2/5 ; 1/5)

N = 4

f(4) = 3n + 1 = 13 f(13) = 3n + 1 = 40 / 2 = 20 / 2 = 10 / 2 = 5

(5 is parent of 13)

f(4) = 4n + 1 = 17 f(17) = 3n + 1 = 52 / 2 = 26 / 2 = 13

(13 is parent of 17)

f(4) = 2n – 1 = 7 f(7) = 3n + 1 = 22 / 2 = 11 (11 is parent of 7)

f(4) = 3n – 1 = 11 f(11) = 3n + 1 = 34 / 2 = 17 (17 is parent of 11)

7 x 3 + 1 = 22 / 2 = 11 x 3 + 1 = 34 / 2 = 17 x 3 + 1 = 52 / 2 = 26 / 2 = 13 x 3 + 1 = 40 / 2 = 20 / 2 = 10 / 2 = 5

13 is Great-Grandparent of 7

17 is Grandparent of 7

11 is Parent of 7

N = -8

f(-8) = 4n + 1 = -31 f(-31) = 3n + 1 = -92 / 2 = -46 / 2 = -23

-23 is parent of -31 (-31 is Grandchild of -17)

f(-8) = 3n + 1 = -23 f(-23) = 3n + 1 = -68 / 2 = -34 / 2 = -17

-17 is parent of -23 (-23 is child of -17)

f(-8) = 2n – 1 = -17 f(-17) = 3n + 1 = -50 / 2 = -25

-25 is Parent of -17 (-25 is parent of -17)

f(-8) = 3n – 1 = -25 f(-25) = 3n + 1 = -74 / 2 = -37

-37 is Parent of -25 (-37 is Grandparent of -17)

-31 x 3 + 1 = -92 / 2 = -46 / 2 = -23 x 3 + 1 = -68 / 2 = -34 / 2 = -17 x 3 + 1 = -50 / 2 = -25 x 3 + 1 = -74 / 2 = -37

N = -6 N = -8

f(-6) = 4n + 1 = -23 f(-8) = 3n + 1 = -23

-23 is Child of -17 -23 is Child of -17

f(-6) = 3n + 1 = -17 f(-8) = 2n – 1 = -17

(3n -1) – (2n -1) = 0 and n = 0 (4n + 1) – (3n + 1) = 0 and n = 0

-1 – -1 = 0 1 – 1 = 0

n = -1 n = 1

-2 4

-1 2

-2 1

-1 4

2

1

(3n -1) – (2n -1) = -2 and n = -2 (4n + 1) – (3n + 1) = -2 and n = -2

-7 – -5 = -2 -7 – -5 = -2

n = -5 n = -7

-14 -20

-7 -10

-20 -5

-10 -14

-5 -7

-14 -20

-7 -10

-20 -5

-10 -14

-5 -7

(3n -1) – (2n -1) = -8 (4n + 1) – (3n + 1) = -8 (4n + 1) – (3n + 1) = -6

and n = -8 and n = -8 and n = -6

-25 – -17 = -8 -31 – -23 = -8 -23 – -17 = -6

n = -31 n = -31 n = -31

-92 -92 -92

-46 -46 -46

-23 -23 -23

-68 -68 -68

-34 -34 -34

-17 -17 -17

-50 -50 -50

-25 -25 -25

-74 -74 -74

-37 -37 -37

-110 -110 -110

-55 -55 -55

-164 -164 -164

-82 -82 -82

-41 -41 -41

-122 -122 -122

-61 -61 -61

-182 -182 -182

-91 -91 -91

-272 -272 -272

-136 -136 -136

-68 -68 -68

-34 -34 -34

-17 -17 -17

MISCELLANEOUS:

*f(-0,5) = 3 x -0,5 + 1 = -0,5 (shortest loop).

*1 is the only positive integer that fulfill three different roles namely Parent, Firstborn Child and Firstborn Grandchild as well and thus ends in a vicious circle (1-4-2-1 loop).

*If it can be proven that the existing loops mentioned above (1 ; 1), (-1 ; -1), (-5 ; -7, -7 ; -5) and (-23 ; -17) are the only 4 anomalies related to the Family Tree Problem, f(n) = 3n + 1, then the Collatz Conjecture is true.

20 October, 2021 at 12:06 pm

AnonymousAny (valid) proof of the Collatz conjecture must use new(!) methods.

21 October, 2021 at 10:06 pm

Gaurav KrishnaHere is the proof for Collatz Conjecture

https://vixra.org/abs/2110.0119

#Collatz #3n+1

27 November, 2021 at 1:18 pm

Anonymouscan I get a good email contact for someone who has studied this conjecture , I think I might have an idea.

1 December, 2021 at 6:37 am

sacirisiWith the Collatz algorithm it is not possible to process all natural numbers because we do not know: quantities and values of even and odd numbers and all their factors. From Tartaglia’s triangle we can detect odd numbers which are the sum of the results of the infinite powers of 2 which have an even index and which are also equal to the previous odd * 4 + 1. These are all the odd numbers that * 3 + 1 generate an even number that is the result of a base power 2 and even index 2 ^ (2 * n≥1) and that, the nth half, ends at 1 because ½ of 2 ^ 1 = 2 ^ 0 = 1. https://www.facebook.com/groups/5349868060/my_posted_content

29 December, 2021 at 7:23 pm

AnonymousCollatz value-dropping formulas:

If the first lower value for odd `c2` is (odd or even) `c3`,

and it takes `p2` halvings and `p3` triplings to get there,

then any (2^`p2`*i + `c2`) has (3^`p3`*i +`c3`) (i >= 0)

as its first lower value,

and `p2` = floor(1+`p3`*log(3)/log(2)) = floor(1+log(3^`p3`)/log(2))

and `p3` = floor(0+`p2`*log(2)/log(3)) = floor(0+log(2^`p2`)/log(3)).

First four formulas, for 1,3,7,11:

(4*i + 1) -> (3*i + 1)

(16*i + 3) -> (9*i + 2)

(128*i + 7) -> (81*i + 5)

(32*i + 11) -> (27*i + 10)

No formula for (4*i+5), (4*i+9), etc., because that is already covered by (4*i+1).

Assuming that for any odd `c2` such a formula can be derived

(notice that this just might! need Collatz to be true),

then Collatz would be proven.

Formula for 27:

((2^59)*i + 27) -> ((3^37)*i + 23)

(so the “sieving power” of 27 is relatively small)

There is no specific reason to stop at the first lower value,

so if a next lower value for any `c2`

can be reached with less assumptions,

that would be stronger.

All even values get halved, so that weeds out 50% of all x > 0

(strikingly similar to how 2 makes all other evens not to be prime).

The formula for (4*i+1) weeds out 2/4th of the odds, so 75% done.

Next formula (16*i + 3) weeds out 2/16th of the odds, which makes it 81%.

Etc. This is similar to Tao’s result that the portion still to go can be made as minimal as desired (but not zero yet).

29 December, 2021 at 8:24 pm

druud62Finally managed to login. Let me know where my wording is overly dramatic etc. :)

29 December, 2021 at 8:50 pm

druud62“No formula for (4*i+5), (4*i+9), etc., because that is already covered by (4*i+1).” is meant to imply that you can do the same for any early formula, so (16*i + 3) -> (9*i + 2) also covers ((16*(i+1)+3) -> (9*(i+1)+2)) = ((16*i+19) -> (9*i+11)), etc.

30 December, 2021 at 12:30 am

satyrcharyHi Dr.TT and others :) I’d appreciate your thoughts on this: https://bit.ly/CollatzProof

Thank you!

30 December, 2021 at 12:59 am

druud62Formula-0 is ((2*i+0)->(1*i+0)), which covers the evens.

https://oeis.org/A122458 is `p3` (with a(0)=1).

The `p2` sequence (2,4,16,4,128,4,32,…) or rather (1,2,4,2,7,2,5,…) can be derived as mentioned from `p3`, and is also A260590 (with a(0)=1 prepended).

30 December, 2021 at 1:06 am

druud62Oh, and please forgive me for using the ‘Collatz proof’ term (read it again and see how I worded it), as that was purely done to trigger some hives.

30 December, 2021 at 3:58 am

druud62For PARI-code, see

https://pari.math.u-bordeaux.fr/archives/pari-users-2112/msg00061.html

Example results:

? run2(2^11+1)

– – – – –

#s : 8197 (bits)

max(c2) : 4099

next(c2): 8245

sieved : 97.24%

took : 2.98s

? run2(2^12+1)

– – – – –

#s : 16389 (bits)

max(c2) : 8195

next(c2): 16437

sieved : 97.76%

took : 12.18s

30 December, 2021 at 6:47 am

druud62Am now checking my sieve-stats once more …

(it’s good to keep doubting)

30 December, 2021 at 8:06 am

druud62All looks fine to me, beyond an irrelevant (…) off-by-factor-2 number in the stats-print.

30 December, 2021 at 8:52 am

druud62If you know of articles about Collatz-sieving, please mention them.

There is also (for example) 8x+5 -> 2x+1 (repeatedly, A350091), and A096773 (one of my favorites).

30 December, 2021 at 3:35 pm

Michael M. RossThere are three nth-term formulas for the whole 3x+1 odd-only map. Two that are linear (A004767, A017077) are sufficient, but be willing to permit n == 0 mod 3. One little trick you’ll have to figure out. The third is a geometric series (A002450) with an adaptation – only if you’re willing to permit (n-1)1/4. Again, one little trick you’ll have to figure out. In fact, defining the N superset with these additions is entirely logical if you make the cognitive switch from even-odd to odd-only. Put the three together and you have one result: Global finitude with respect to 1.

30 December, 2021 at 4:10 pm

druud62The next trick I am thinking about, is to use a repeated transformation from the odd towards its first lower value (either even or odd would be fine) that has a built-in switch to “degrade” into divisions-by-1/additions-of-0 once the lower value is reached.

30 December, 2021 at 4:32 pm

druud62Odd-only is fine, and is mostly how I reason about it. But it is like calling 2 a special prime only because it is not odd, to which I never hesitate to object. To me 2 is special because it controls exactly 50%. :)

30 December, 2021 at 7:37 pm

Michael M. RossBut I ask you, if you know the nth term of any sequence, does that not mean the game is over for this conjecture? It’s not a dynamical system – it’s a deterministic, well-ordered system. I’ve known this from a computational framework for a long time, but it might be more meaningful to mathematicians to have an nth-term solution. We’ll see. https://qr.ae/pG6p8y. As I see it, this conjecture is not an isolated, rugged mountain, more a playground in the garden of mathematics.

30 December, 2021 at 8:17 pm

Aditya Guha RoyI would request you all to stop spamming this space. More than 70% of the comments on page 4 are rubbish and one should realize that this is a bad thing. If you have genuine ideas please post them on arxiv or send it to some appropriate journal. Like any other mathematician, prof. Tao is already too busy with many important things, and so it is not at all rational to assume that he would go over any of these computational algorithms. It is just his generosity that he has not deleted these comments or that commenting on his blog doesn’t require the wordpress approval feature. Respect this thing but don’t take it as your impunity to spam here. Many important comments may go unnoticed this way.

30 December, 2021 at 8:26 pm

satyrcharyAditya, good point. I’d LOVE to delete the only comment I put up above (12:30am) – but am unable :(

I did not look before posting, about being able to delete (simply assumed I could!).

31 December, 2021 at 12:44 am

druud62Yes, it would good to be able to (delete your own) comments.

31 December, 2021 at 2:56 am

Alberto IbañezFor every odd n, exist infinite solutions ?

20 August, 2022 at 7:39 pm

AndersaysI have a much simpler method that also works for the negatives. To see a glimpse, a 3-column spreadsheet is all that is needed. First column (C1) is just a list of integers from 0 to n (any arbitrary number will do, 100 is usually a nice amount). Second column (C2) is the Collatz using C1 as the value for x in 2x+1. Third column (C3) is the mod(C2,6) command. Simple predictability from a much simpler approach.