Iteration Steps of 3x+1 Problem

Youchun Luo

REVIEW 3 major objections 5 minor 10 references

Iteration Steps of 3x+1 Problem

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper shows that, conditional on the Collatz conjecture and a proposed Weak Residue Conjecture, the total, odd, and even step counts of any trajectory are forced to satisfy six linked floor-and-ceiling formulas.

desk verdict The central theorem is the Weak Residue Conjecture rewritten; the only new result is a weak bound resting on an unchecked finite calculation. read the letter →

arxiv 2506.23070 v3 pith:BBECBIAR submitted 2025-06-29 math.GM

Youchun Luo This is my paper · ORCID

classification math.GM

keywords Collatzconjecture3x+1problemresidueWeaktotalstoppingtimeiterationstepslogarithmicformulasoddandevenstepcounts

topics The Collatz Conjecture

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

On the 3x+1 problem, every trajectory that reaches 1 has three basic statistics: total steps $D(N)$, odd steps $O(N)$, and even steps $E(N)$. This paper tries to show that these three numbers are not independent: it introduces a Weak Residue Conjecture stating that the residue $2^{E(N)}/(3^{O(N)}N)$ never exceeds 2, and it proves that if both that conjecture and the 3x+1 conjecture are true, six exact floor-and-ceiling formulas with logarithms to bases 6, 2, and 3 relate $D(N)$, $O(N)$, and $E(N)$. The payoff is that knowing any one of the three counts, together with $N$, determines the other two, so the parity composition of a trajectory becomes a rigid arithmetic constraint rather than a random feature. The paper also proves a conditional upper bound on the residue that confirms the Weak Residue Conjecture for trajectories with at most 512 odd steps.

What carries the argument

The carrying object is the residue, defined by $\mathrm{Res}(N)=2^{E(N)}/(3^{O(N)}N)$. The proof runs through the interval $1\le \mathrm{Res}(N)\le 2$: taking base-2 logarithms converts that interval into inequalities such as $0\le D(N)-\log_2 6\cdot O(N)-\log_2 N\le 1$, and because $D(N)$, $O(N)$, $E(N)$ are integers, each such width-one inequality has exactly one admissible integer, which the floor-and-ceiling formulas select. For the upper bound $\mathrm{Res}(N)<O(N)^{1/9}$, the paper uses the product formula $\mathrm{Res}(N)=\prod_j (1+1/(3N_j))$ over the odd values in the trajectory, together with the fact that all odd values after the first are not divisible by 3, which reduces the estimate to a harmonic sum over the smallest admissible odd integers.

What would settle it

Find an integer $N$ whose Collatz trajectory reaches 1 and has more than 512 odd steps, then compute $\mathrm{Res}(N)=2^{E(N)}/(3^{O(N)}N)$; any value above 2 would refute the Weak Residue Conjecture and thereby the premise of the main theorem. The paper's own corollary guarantees no such counterexample exists with $O(N)\le 512$, so the decisive search region is the high-odd-step regime.

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Extended reading notes

Core claim

The central claim is a conditional structural discovery: assuming both the 3x+1 conjecture and the Weak Residue Conjecture $\mathrm{Res}(N)\le 2$, the paper derives six identities, beginning with $O(N)=\lfloor \log_6 2\,D(N)-\log_6 N\rfloor$ and $E(N)=\lceil \log_6 3\,D(N)+\log_6 N\rceil$, with four analogous formulas obtained by using $D$, $O$, or $E$ as the known quantity. The derivation starts from $1\le \mathrm{Res}(N)\le 2$, takes logarithms, and produces a width-one interval that contains exactly one integer, so the floor or ceiling is forced. The paper further proves, under only the 3x+1 assumption, that $\mathrm{Res}(N)<O(N)^{1/9}$ for $O(N)\ge 20$, and hence $\mathrm{Res}(N)<2$ whenever $O(N)\le 512$; this supports the Weak Residue Conjecture without completing its proof.

Load-bearing premise

The theorem rests on the new Weak Residue Conjecture — that $2^{E(N)}/(3^{O(N)}N)$ never exceeds 2 — together with the 3x+1 conjecture itself, and the paper only proves this residue bound for trajectories with at most 512 odd steps; if either assumption fails, the six formulas are not established.

Editorial extensions

If this is right

If both conjectures hold, then for every $N$ whose trajectory reaches 1 the values of $O(N)$ and $E(N)$ are fixed by $D(N)$ and $N$ alone, so the odd/even split can be read off without counting parities along the trajectory.
The six formulas are mutually redundant: any one of them, together with $D(N)=O(N)+E(N)$, forces the other five, so the whole system is one relation in three integer variables.
Because the paper proves $\mathrm{Res}(N)<2$ for $O(N)\le 512$, the formulas are already secured across that entire range (conditional on the 3x+1 conjecture); the unproved residue bound only matters for trajectories with more than 512 odd steps.
Formula (15), previously proposed in the literature as a standalone conjecture, is shown to be one component of a six-formula system, giving the earlier conjecture a broader context.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: because $\log_2 \mathrm{Res}(N)$ is exactly the gap between $D(N)$ and $\log_2 6\,O(N)+\log_2 N$, each formula is logically equivalent to the Weak Residue Conjecture; the real open problem behind the theorem is proving that residue bound, not the algebra that follows from it.
Beyond the paper: a numerical search concentrating on trajectories with $O(N)>512$ would directly test the Weak Residue Conjecture, and any trajectory with residue above 2 would destroy the premise on which the six formulas rest.
Beyond the paper: the same argument should carry over to the $qx+1$ generalization announced in the abstract, with the base 6 replaced by $2q$; checking this would show whether the mechanism is specifically tied to the constant 3 or is a general feature of such iterations.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the paper's main theorem is the Weak Residue Conjecture in logarithmic disguise, so the advertised computability of O and E from D is not new. The only genuinely new item, the O(N)^(1/9) bound for Res(N), is a weak partial step and rests on an unverified finite check.

The paper is honestly written: it credits Ruggiero for Eq. (15), states WRC as a conjecture, and gives a clean alternate proof of Res(N) ≥ 1. The derivation of the six formulas from 1 ≤ Res ≤ 2 is algebraically correct. But that interval assumption is exactly WRC; substituting Res = 2^E/(3^O N) turns the double inequality into 0 ≤ D − O log2 6 − log2 N ≤ 1, which is what Eq. (15) says. So Theorem 3 is conditional on an unproven statement that is equivalent to its conclusion. That's not a circular proof, but it means the headline claim—compute O and E from D—is hollow until WRC is settled.

Theorem 2's bound Res(N) < O(N)^(1/9) for O(N) ≥ 20 is the real content. It's a clever use of the smallest odd non-multiples of 3, and the numerical constant is tight. But the bound gives Res < 2 only for O ≤ 512, far from the WRC range, and the verification for 20 ≤ O ≤ 1252 is asserted without code or a table, so it's not reproducible as written. There are also some notation slips, and the abstract promises a qx+1 analysis that doesn't appear in the text.

I wouldn't send this to a serious referee. The logical structure is fine but the central result is a restatement, and the new bound is too slight and incomplete to justify peer-review time. If the finite check is supplied and the author wants to publish the bound as a small note, that might work in an informal venue. For now, I'd treat it as an honest but substantially empty attempt.

Referee Report

3 major / 5 minor

Summary. The manuscript studies the Collatz residue Res(N) = 2^{E(N)} / (3^{O(N)} N), where O(N) and E(N) are the numbers of odd and even steps in the 3x+1 trajectory of N until 1. It proves Res(N) ≥ 1 under the 3x+1 conjecture (Theorem 1), and claims an upper bound Res(N) < O(N)^{1/9} for O(N) ≥ 20 (Theorem 2), relying on an asserted finite check. The main result (Theorem 3) states that, assuming the 3x+1 conjecture and the Weak Residue Conjecture (WRC), Res(N) ≤ 2, the six quantities D, O, E are linked by floor/ceiling formulas, so that any one determines the other two. The abstract also announces a qx+1 generalization. The paper's verification evidence is limited to small O(N), and the central theorem is conditional on the unproved WRC.

Significance. If the Weak Residue Conjecture were proved, the formulas in Theorem 3 would be a compact way to compute odd/even step counts from total stopping time, and would strengthen known results on the structure of Collatz trajectories. The paper deserves credit for a self-contained proof of Res(N) ≥ 1 via the product formula in Lemma 1, and for attempting an analytic upper bound in Theorem 2. However, as stated, the paper does not establish WRC; its main theorem assumes WRC and derives formulas that are logically equivalent to WRC after taking logarithms. The partial bound in Theorem 2 is incomplete because of the unverified finite range, and even complete would not cover the O(N) > 512 regime. Consequently the manuscript's principal new claim is a reformulation of an unproved conjecture, not a theorem with independent content.

major comments (3)

[Section 4, Theorem 3, Eq. (15)] Equation (15) is WRC in logarithmic form. The proof of Eq. (15) starts with 1 ≤ Res(N) ≤ 2. Since log2 Res(N) = D(N) - log2 6·O(N) - log2 N, the inequality 1 ≤ Res(N) ≤ 2 is exactly Eq. (19): 0 ≤ D(N) - log2 6·O(N) - log2 N ≤ 1. The subsequent ceiling argument only converts this double inequality into the equivalent integer statement D(N) = ⌈log2 6·O(N) + log2 N⌉. Thus Eq. (15) is not derived from WRC; it is WRC rewritten. The same equivalence holds for Eqs. (13), (14), (16), (17), (18), all of which are rearrangements of the same double inequality. Therefore Theorem 3 does not provide new evidence for WRC, and the advertised computability of O and E from D is exactly the content of the conjecture the paper assumes.
[Section 3, Theorem 2 and Corollary 1] The theorem's claimed upper bound rests on an unshown finite verification for 20 ≤ O(N) ≤ 1252. The sentence 'We can verify that...' is not a proof: no table of maximum values, no algorithm, and no code is supplied. This is a load-bearing gap because the analytic steps cover only O(N) ≥ 1253 (and the excluded residue classes are handled only for a few exceptional N). Moreover, the conclusion Res(N) < O(N)^{1/9} cannot support the WRC for O(N) > 512, since O(N)^{1/9} > 2 in that range. Corollary 1 is the only part of Theorem 2 that yields Res(N) < 2, and it is restricted to O(N) ≤ 512.
[Abstract and body scope] The abstract states that the qx+1 problem is studied in the same way, but the manuscript text contains no section or argument devoted to qx+1. Either the qx+1 part has been omitted in the submitted version, or the abstract overstates the scope. In either case, the paper as written is incomplete with respect to its advertised contents.

minor comments (5)

[Eq. (2)] Equation (2) identifies E(N) with the total stopping time σ∞(N); the standard total stopping time is D(N), as used in Eq. (3) and the example N=7. Please correct this equation or clarify the definition.
[Section 3] The notation 'N1 = 0 (mod 3)' should be 'N1 ≡ 0 (mod 3)' to avoid confusion with equality.
[Introduction] The finite verification claims in the introduction for intervals near 10^20, 10^50, 10^200 are not reproducible; they should be flagged as numerical observations or accompanied by code, consistent with the journal's standards.
[Lemma 2] The proof of Lemma 2 contains an incomplete sentence fragment: 'obviously Nk−1 is not an integer, it's a contradiction .' This should be reworded.
[Section 4, after Eq. (15)] The paper mentions that Eq. (15) was conjectured by Ruggiero [10] but does not discuss the relationship; since Eq. (15) is equivalent to WRC, the connection to [10] merits a brief comment in Section 4.

Circularity Check

2 steps flagged · score 8.0 of 10

Theorem 3's six formulas are the Weak Residue Conjecture restated in logarithmic form; the derivation substitutes WRC into the definition of Res and solves the same inequality back out.

self definitional [Section 4, Theorem 3, proof of Eq. (13), page 13]
"Equation (13): Since 1 ≤ Res (N ) ≤ 2, 1 ≤ 2E(N ) 3O(N )·N = 2D(N )−O(N ) 3O(N )·N ≤ 2, using logarithm function, we have 0 ≤ D(N ) − log2 6 · O(N ) − log2 N ≤ 1 (19)"

By the paper's own definitions, Res(N)=2^{E(N)}/(3^{O(N)}N) (Eq. 4) and E(N)=D(N)-O(N) (Eq. 3), so log2 Res(N) = D(N) - O(N) - O(N) log2 3 - log2 N = D(N) - O(N) log2 6 - log2 N. Therefore the double inequality (19) is exactly log2(1) ≤ log2 Res(N) ≤ log2(2), i.e. 1 ≤ Res(N) ≤ 2. The lower bound is proved independently in Theorem 1, but the upper bound is precisely the WRC proposed in Eq. (5). Solving (19) for O(N) gives Eq. (13); rearranging (19) for D(N) gives Eq. (15). Thus the derived formulas are the conjecture's own inequality in logarithmic notation, not a consequence of WRC plus independent facts. Theorem 2 only gives Res(N) < O(N)^{1/9}, which exceeds 2 as soon as O(N)>512, so it does not supply the missing upper bound.
self definitional [Section 4, Theorem 3, proof of Eq. (16), page 14]
"Equation (16): Since 1 ≤ Res (N ) ≤ 2, 1 ≤ 2E(N ) 3O(N )·N ≤ 2, using logarithm function, we have 0 ≤ E(N ) − log2 3 · O(N ) − log2 N ≤ 1 (21)"

The proof of Eq. (16) again starts from '1 ≤ Res(N) ≤ 2' and writes Res(N)=2^{E(N)}/(3^{O(N)}N), obtaining 0 ≤ E(N) - log2 3·O(N) - log2 N ≤ 1 (21). Since log2 Res(N) equals the left-hand expression, (21) is WRC in another logarithmic form. Eq. (16) and Eq. (18) are the same inequality solved for E(N) and O(N); Eq. (14) and Eq. (17) play the same game with E(N)=D(N)-O(N). None of these formulas adds content beyond the assumed WRC plus the already-proved lower bound. The 'nontrivial relationships' advertised in the abstract are therefore the input conjecture rearranged.

full rationale

The paper is formally honest in stating Theorem 3 as a conditional: if the 3x+1 conjecture and the Weak Residue Conjecture are true, then six equations connect D, O, and E. However, the derivation is not an independent derivation of those equations. The proof begins with the WRC inequality 1 ≤ Res(N) ≤ 2, substitutes the paper's own definitions Res(N)=2^E/(3^O N) and E=D-O, and obtains exactly 0 ≤ D - log2 6·O - log2 N ≤ 1. That double inequality is WRC in logarithmic form. All six formulas are obtained by rearranging this same inequality and applying floor or ceiling to isolate one of D, O, E. Equation (15) is even acknowledged to have been conjectured before by Ruggiero (2019). The only independent ingredient is the lower bound Res(N) ≥ 1, proved in Section 2; the upper bound Res(N) ≤ 2 is never proved. Theorem 2 gives Res(N) < O(N)^{1/9}, which implies Res(N) < 2 only when O(N) ≤ 512; for larger O(N) the bound exceeds 2, so WRC remains unsupported. The finite check cited for 20 ≤ O(N) ≤ 1252 is asserted without reproducible table or code. Thus the central advertised result reduces, by definition and algebraic rearrangement, to an unproved conjecture. Score 8 reflects that the claimed relationships are forced by the assumed WRC itself, not by self-citation; the lower-bound half of the inequality is independently established, but the load-bearing upper bound is the conjecture rewritten.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper's central theorem rests on two unproven conjectures, one of which (WRC) is introduced by the paper itself and is logically equivalent to one of the derived equations. The finite verification is an additional unsupported assertion. There are no fitted free parameters or invented entities.

assumptions (3)

domain assumption 3x+1 (Collatz) conjecture
Assumed in Lemma 1, Theorem 1, Theorem 2, Corollary 1, and Theorem 3; the quantities D, O, E and Res are defined only for trajectories that reach 1.
ad hoc to paper Weak Residue Conjecture (Res(N) <= 2)
Proposed as Eq. (5) in Section 1; it is the premise of Theorem 3, and Eq. (15) is essentially a restatement of it.
ad hoc to paper Finite verification for 20 <= O(N) <= 1252 in Theorem 2
The proof of Theorem 2 asserts a numerical check over the interval O(N) in [20, 1252] without providing code or details; the claimed upper bound depends on this check.

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Cite this review

Pith. "Pith review of Iteration Steps of 3x+1 Problem." pith.science (2026). https://pith.science/paper/BBECBIAR

@misc{pith2026250623070,
  author       = {Pith},
  title        = {Pith review of: Iteration Steps of 3x+1 Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BBECBIAR}},
  note         = {Machine review of arXiv:2506.23070}
}

abstract

On the 3x+1 problem, given a positive integer $N$, let $D\left( N \right) $, $O\left( N \right) $ and $E\left( N \right) $ denote the total number of iteration steps, the number of odd iteration steps, and the number of even iteration steps, respectively, when $N$ is iterated until it reaches 1. It is straightforward to observe that $D\left( N \right) =O\left( N \right) +E\left( N \right) $. In this paper, we propose a conjecture termed the Weak Residue Conjecture(i.e., $\frac{2^{E\left( N \right)}}{3^{O\left( N \right)}\cdot N}<2$). We prove that if the 3x+1 conjecture is true and the Weak Residue Conjecture is true, there exist nontrivial relationships among $D\left( N \right) $, $O\left( N \right) $, $E\left( N \right) $, i.e., $O\left( N \right) =\lfloor \log _62\cdot D\left( N \right) -\log _6N \rfloor $(this implies that, given $N$, both $O\left( N \right) $ and $E\left( N \right) $ can be directly computed from $D\left( N \right) $), and five more similar equations are derived simultaneously. Similarly, the case of qx+1 problem is studied too.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

Lagarias, The 3x+1 problem and its generalizations, Amer

Jeffrey C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92(1):3-23, 1985

work page 1985
[2]

The 3x+1 Problem: An Annotated Bibliography, II (2000-2009)

Jeffrey C. Lagarias, The 3x+1 Problem: An Annotated Bibliography, II (2000-2009), arXiv:math/0608208, 2012

work page Pith review arXiv 2000
[3]

Tom´ as Oliveira e Silva, Empirical verification of the 3x+1 and re- lated conjectures, The ultimate challenge: the 3x+1 problem, American Mathematical Society (AMS) (ISBN 978-0-8218-4940-8/hbk), 189-207, 2010. 15

work page 2010
[4]

Leavens, Mike Vermeulen, 3x+1 search programs, Computers and Mathematics with Applications, 24(11):79-99, 1992

Gary T. Leavens, Mike Vermeulen, 3x+1 search programs, Computers and Mathematics with Applications, 24(11):79-99, 1992

work page 1992
[5]

Paul Andaloro, On total stopping times under 3x + 1 iteration, Fi- bonacci Quart, 38(1):73-78, 2000

work page 2000
[6]

Ladue, Clusters of Integers With Equal Total Stopping Times in the 3X + 1 Problem, Fibonacci Quart, 56(2):156-162, 2018

Mark D. Ladue, Clusters of Integers With Equal Total Stopping Times in the 3X + 1 Problem, Fibonacci Quart, 56(2):156-162, 2018

work page 2018
[7]

Eric Roosendaal, On the 3x + 1 problem, https://www.ericr.nl/wondrous/, 2025(Accessed: 2025-6-3)

work page 2025
[8]

Stefan Andrei, Manfred Kudlek, Radu Stefan Niculescu, Some results on the Collatz problem, Acta Informatica, 37(2):145-160, 2000

work page 2000

Show all 10 references

[9]

Monks, 3x+1 Minus the +, Discrete Mathematics and The- oretical Computer Science, 5(1):47–54, 2002

Kenneth G. Monks, 3x+1 Minus the +, Discrete Mathematics and The- oretical Computer Science, 5(1):47–54, 2002

2002
[10]

Rafael Ruggiero, The relationship between stopping time and number of odd terms in Collatz sequences, arXiv:1911.01229, 2019. 16

1911 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Lagarias, The 3x+1 problem and its generalizations, Amer

Jeffrey C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92(1):3-23, 1985

work page 1985

[2] [2]

The 3x+1 Problem: An Annotated Bibliography, II (2000-2009)

Jeffrey C. Lagarias, The 3x+1 Problem: An Annotated Bibliography, II (2000-2009), arXiv:math/0608208, 2012

work page Pith review arXiv 2000

[3] [3]

Tom´ as Oliveira e Silva, Empirical verification of the 3x+1 and re- lated conjectures, The ultimate challenge: the 3x+1 problem, American Mathematical Society (AMS) (ISBN 978-0-8218-4940-8/hbk), 189-207, 2010. 15

work page 2010

[4] [4]

Leavens, Mike Vermeulen, 3x+1 search programs, Computers and Mathematics with Applications, 24(11):79-99, 1992

Gary T. Leavens, Mike Vermeulen, 3x+1 search programs, Computers and Mathematics with Applications, 24(11):79-99, 1992

work page 1992

[5] [5]

Paul Andaloro, On total stopping times under 3x + 1 iteration, Fi- bonacci Quart, 38(1):73-78, 2000

work page 2000

[6] [6]

Ladue, Clusters of Integers With Equal Total Stopping Times in the 3X + 1 Problem, Fibonacci Quart, 56(2):156-162, 2018

Mark D. Ladue, Clusters of Integers With Equal Total Stopping Times in the 3X + 1 Problem, Fibonacci Quart, 56(2):156-162, 2018

work page 2018

[7] [7]

Eric Roosendaal, On the 3x + 1 problem, https://www.ericr.nl/wondrous/, 2025(Accessed: 2025-6-3)

work page 2025

[8] [8]

Stefan Andrei, Manfred Kudlek, Radu Stefan Niculescu, Some results on the Collatz problem, Acta Informatica, 37(2):145-160, 2000

work page 2000