How Prime Factors Form Fractals

Micah D. Tillman

REVIEW 2 major objections 4 minor 17 references

How Prime Factors Form Fractals

T0 review · 2 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read This paper claims that the exponents of a fixed prime in the integers form fractal sequences, and that the 2-adic valuation sequence draws the Lévy Dragon.

desk verdict A clean, honest synthesis of known p-adic valuation patterns with dragon curves; the discrete math is solid, but the geometric conclusion outruns the proof. read the letter →

arxiv 2502.15743 v1 pith:LRCMTIXP submitted 2025-02-08 math.GM

Micah D. Tillman This is my paper · ORCID

classification math.GM MSC 11A5128A8011B83

keywords p-adicvaluationfractalsequenceprimefactorizationsieveLévyDragonHeighwayself-similarsequencesoddpartofndivision-free

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

The paper claims that the apparent chaos of prime factorizations hides a repeatable fractal structure. It builds a division-free sieve—an algorithm that repeatedly copies, concatenates, and increments a seed sequence of zeros—and proves that the row for each prime p records exactly the exponent of p in each integer n. It then proves that every such p-adic valuation sequence is a fractal sequence in a defined sense: predictably self-containing and aperiodic. Finally, it claims that reading the 2-adic valuation sequence as a number of 90-degree turns draws the Lévy Dragon, and that the odd part of n sequence draws the Heighway Dragon. A sympathetic reader would take away that prime factorization carries geometric fractal structure.

What carries the argument

The engine is the duplicate-concatenate-increment rule. For each prime p, begin with the one-term sequence ⟨0⟩; at each stage make p−1 copies of the current sequence, concatenate them after it, and add 1 to the final term. The nth entry of the resulting sequence is the exponent of p in n, so the sieve's rows are exactly the p-adic valuations. Fractality is certified by a decimation rule that selects indices that are multiples of p+1; because p never divides p+1, the term at f(p+1) equals the term at f, giving a predictable copy of the sequence inside itself. For the Lévy Dragon, the matching object is the turn algorithm that starts with ⟨3⟩, increments every entry, inserts a new 3 between adjacent entries, and appends 3s at both ends; Theorem 15 shows this sequence is identical to the 2-adic valuation sequence at indices that are multiples of 8.

What would settle it

Compute the turtle path for the 2-adic valuation sequence at 90-degree turns over the first 8^N indices and measure its Hausdorff distance to the Lévy Dragon polyline after N iterations; if that distance does not tend to 0 as N grows, the paper's convergence claim fails even though the turn sequence matches.

Watch

Extended reading notes

Core claim

The paper's central discovery is that the exponents of any fixed prime p across the positive integers—the p-adic valuation sequence—form a fractal sequence, and that for p=2 the same sequence, drawn as a path that turns v2(n) quarter-turns at step n, produces the Lévy Dragon. The sieve that reaches this conclusion starts from a single 0 and repeatedly expands it by making p−1 copies, appending them, and increasing the final term by 1; Theorem 6 shows this reproduces prime factorizations without division. Lemma 12 exhibits a simple decimation rule—take every (p+1)-st term—that locates a copy of the whole sequence inside itself, and Lemma 13 shows no periodic block generates it. Theorem 15 proves that the terms at indices divisible by 8 exactly match an algorithm for the Lévy Dragon's turn sequence, with the intervening terms forming small 'T' detours. In the appendix, the same construction is carried for the odd part of n, whose values modulo 4 match the Heighway Dragon's turns.

Load-bearing premise

The geometric conclusion rests on the unproven claim that the 'T'-shaped detours in the v2 curve become negligible relative to the whole figure as more indices are added.

Editorial extensions

If this is right

A student can generate prime factorizations by mechanical copying rather than division, making the exponent pattern visible.
Every prime's exponent sequence is predictably self-containing and aperiodic, so each prime contributes its own fractal layer to the natural numbers.
The 2-adic valuation sequence encodes the Lévy Dragon's turns at every index divisible by 8; the intervening 'T' shapes are the only difference from the standard construction.
The odd part of n, taken modulo 4, reproduces the Heighway Dragon's turn sequence, tying the other classical dragon curve to integer factorization.
The paper's open questions point to a family of angle-and-prime combinations that generate additional fractal figures, and to a possible L-system translation of the sieve.

Reading between the lines

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

I infer that if the spike-shrinking claim is formalized in a Hausdorff-metric or length-ratio sense, the v2 curve converges to the Lévy Dragon exactly; the paper states the tendency but leaves the metric unspecified.
The duplicate-concatenate-increment rule is morphic in flavor, so a natural extension is a string-replacement proof that p-adic valuation sequences are automatic or morphic words, connecting them to the paperfolding literature.
The matching at multiples of 8 may yield an explicit formula: the Lévy turn sequence is the 2-adic valuation of 8n, which could simplify dragon-curve turn computations.

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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

Colleague,

This is a likeable paper, but its headline claim is not fully proved. The discrete core is sound: the sieve does generate the p-adic valuations, and the self-containment at indices p+1 plus unboundedness gives aperiodicity under the paper's own definition of 'fractal sequence.' Theorem 15 is the most concrete new observation: the Lévy turn-insertion algorithm exactly produces v2 at multiples of 8. That is a clean, checkable identity.

The weak point is Section 4.5. The paper only matches the turn sequence at the 'spikes' (multiples of 8). Between those, v2 contains the fixed block 0,1,0,2,0,1,0, which makes a bounded rectangular excursion. The paper asserts that this excursion becomes negligible relative to the whole curve as iterations grow, but it never gives a metric or a convergence argument. Without that, Theorem 15 gives you a matching subsequence of turns, not a proof that the full v2 curve approaches the Lévy Dragon. The stress-test note is right about this.

The Heighway Dragon appendix is on firmer ground. There, the connection is an exact mod-4 identity between the odd-part sequence and the paperfolding turn sequence, and the insertion algorithms are shown to agree. That part is convincing.

Novelty is modest: the paper credits Cloitre for the sieve construction and cites OEIS entries that already label these sequences fractal and already link A346070 and A099545 to dragon curves. What is new is the unified presentation and the explicit theorem about multiples of 8. No code or machine-checked proofs are shipped, but the discrete claims are easy to verify by hand.

Who is this for? Someone teaching elementary number theory or looking for a visually compelling way to present p-adic valuations. It is not a paper that resolves an open problem, and the geometric section needs more rigor before the central metaphor becomes a theorem. Still, the paper is honest about its own limits and the discrete mathematics is correct.

I would send it to a referee: the discrete results deserve a record, and a referee could push the author to either tighten Section 4.5 or soften the claim. I would not cite it as a proof of the Lévy-Dragon connection, but I would cite Theorem 15 if I needed that specific identity.

Referee Report

2 major / 4 minor

Summary. The paper introduces a table-based sieve that successively generates, for each prime p, the sequence of p-adic valuations v_p(n), and proves (Theorem 6) that the sieve outputs exactly the primes and the prime factorizations. It then defines a fractal sequence as one that is predictably self-containing and aperiodic, proves (Theorem 14) that every v_p sequence is fractal via the decimation rule selecting indices f(p+1) and via unboundedness of the terms, and claims (Section 4) that v_2, read as the number of 90-degree counterclockwise turns at each integer, generates the Lévy Dragon. The main supporting result (Theorem 15) is that the terms of v_2 at indices divisible by 8 follow exactly the algorithm for the Lévy dragon's turn sequence. An appendix relates the odd part of n modulo 4 to the Heighway Dragon turn sequence.

Significance. If fully established, the paper would provide a clean exposition of a known but underappreciated fact: p-adic valuation sequences are self-similar in a precise subsequence sense, and v_2's turn sequence is related to the Lévy Dragon. The discrete portions are largely correct and self-contained, and the paper is honest in not fitting parameters and in comparing against OEIS benchmarks. However, the geometric equivalence in Section 4.5 is asserted rather than proved, and the paper's own Section 5 acknowledges that the other 'fractal figures' are not formalized. The contribution is therefore a useful set of exact identities and a suggestive geometric conjecture, rather than a complete proof of the headline geometric claim.

major comments (2)

[Section 4.5] The sentence 'the size of these deviations, relative to the overall figure, becomes negligible as the number of iterations increases' is the load-bearing step connecting Theorem 15 to the claim that the curve generated by v_2 is the Lévy Dragon, but no metric or convergence theorem is supplied. Theorem 15 proves that the finite turn sequence generated by the Lévy-dragon algorithm at iteration m equals v_2(8i) for i=1..2^m-1; it does not describe the curve generated by all terms of v_2, and the intervening block v_2(8k+1..8k+7)=0,1,0,2,0,1,0 has zero net turn but a bounded nonzero excursion. To establish the claimed limit, the paper needs a precise notion, such as Hausdorff metric on normalized curves or a path-length accounting showing that the extra T-excursions vanish as m tends to infinity, together with a proof that they do so. Without this, the paper proves a turn-subsequence identity, not geometric equivalence of the full v_2 curve.
[Section 4.1] The geometric interpretation of v_2 is introduced informally: terms are said to represent 'the number of 90-degree turns to make at n,' but the formal curve construction is never defined, and the equivalence 'four 90-degree turns are the same as none' is used without proof. Since v_2 contains arbitrarily large terms, the curve is only defined up to congruence modulo 4, and that reduction should be stated explicitly as part of the definition of the map from sequences to curves. This clarification is needed before the equivalence with A346070 or with the Lévy Dragon can be made rigorous.

minor comments (4)

[Section 2.4, Proposition 4] The sentence 'we want p + 1 at that index' should read 'we want j + 1 at that index'; the surrounding argument is otherwise clear.
[Definition 7] There is a duplicated article in 'the the number returned by vp(ni)'; this should be corrected.
[Section 4.5] The phrase 'can be confident that v2⟨⟩ produces the Lévy Dragon' appears to be missing a 'we' and should read 'can we be confident that v2⟨⟩ produces the Lévy Dragon.'
[Theorem 15] The proof's 'leaves no gaps' step is persuasive but informal; stating the result as an explicit induction on the binary length of the index, or equivalently as l_i = v_2(8i) for every positive integer i, would make the argument easier to verify.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the sieve-to-valuations proof, the fractal-sequence theorems, and the dragon-curve identifications are self-contained; the only weak point is an unproved convergence assertion in Section 4.5, which is an evidence gap rather than a circular step.

full rationale

The derivation chain is self-contained and does not reduce to its inputs. Theorem 6 is proved by an explicit induction (Propositions 2 and 4) showing the sieve entries equal v_p(n); no parameter is fitted and no result of the present author is invoked. Theorem 14 proves the two defining properties of the paper's stipulative Definition 8 directly: Lemma 12 uses the decimation rule at indices f(p+1) together with the fact that p does not divide p+1, and Lemma 13 uses the unbounded growth of terms. Calling these sequences 'fractal' under a chosen definition is a definitional choice, not circularity. Theorem 15 derives the Levy-dragon turn algorithm from independent geometric sources (Alster, Riddle, Levy) and then proves equality with the v2 terms at multiples of 8; the Heighway-dragon appendix similarly derives the turn sequence from the paper-folding construction and proves equality with the odd part of n modulo 4 against the independent OEIS benchmark A000265/A099545. The one genuinely weak passage is Section 4.5, where the claim that the 'T' deviations 'become negligible as the number of iterations increases' is asserted without a metric or limit proof; however, that is an unsupported analytic-convergence assertion, not a case of a prediction being forced by construction or of a load-bearing self-citation. Accordingly, no circular step can be exhibited with a quote-and-reduction, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

No free parameters are fitted. The paper introduces no new mathematical objects; it works with p-adic valuations and standard dragon curves. The main assumptions are the paper-specific definition of 'fractal' and the informal geometric convergence.

assumptions (2)

ad hoc to paper The paper's Definition 8: a sequence is fractal if it is predictably self-containing and aperiodic.
This definition is introduced in the paper and differs from Kimberling's standard lower-trim definition of fractal sequences (cited as [8]). The proof of Theorem 14 shows the valuation sequences satisfy this definition, but whether 'fractal' should mean this is an assumption the paper adopts.
domain assumption The identification of the Levy Dragon's turn sequence with the terms of v2 at indexes that are multiples of 8, together with the negligible-spike limit.
The geometric equivalence relies on interpreting integer terms as counts of 90-degree turtle turns, a modeling choice that is not formally justified. The convergence of the spiky curve to the Levy Dragon is stated informally in Section 4.5 and not proved as a limit theorem.

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Cite this review

Pith. "Pith review of How Prime Factors Form Fractals." pith.science (2026). https://pith.science/paper/LRCMTIXP

@misc{pith2026250215743,
  author       = {Pith},
  title        = {Pith review of: How Prime Factors Form Fractals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LRCMTIXP}},
  note         = {Machine review of arXiv:2502.15743}
}

read the original abstract

We explore a new sieve that generates both primes and prime factorizations, without resorting to division. We demonstrate that the integer sequences generated by the sieve are the p-adic valuations of n, and that each is a fractal sequence. We then show that these sequences produce geometrical fractals like the Levy Dragon. We end by showing the connection between the odd part of n integer sequence and the Heighway Dragon.

Figures

Figures reproduced from arXiv: 2502.15743 by the authors.

**Figure 1.** The number line as a “tree.” Presented in this way, the number line becomes something like a rooted tree, where the (equal) length of each edge represents the fact that all edges have a weight of 1. Nothing 12 [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗

**Figure 2.** The number line, with 90 degree turns representing [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** The number line (without numbers) with 90-degree turns representing [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (17 more)

**Figure 4.** Figure 4: A curve with 90-degree turns representing [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 5.** Figure 5: A curve with 90-degree turns representing [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 6.** Figure 6: A “spike” on the L´evy Dragon. In fact, it might be more accurate to say that v2⟨⟩ consists of instances of that subsequence connected by a series of other numbers, each greater than 2. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: The first five iterations of the L´evy Dragon. [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

**Figure 8.** Figure 8: The number line with 90-degree turns representing [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]

**Figure 9.** Figure 9: The number line with 120-degree turns representing [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]

**Figure 10.** Figure 10: The number line with 135-degree turns representing [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]

**Figure 11.** Figure 11: The number line with 120-degree turns representing [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]

**Figure 12.** Figure 12: The number line with 60-degree turns representing [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]

**Figure 13.** Figure 13: A “strip of paper” to be folded into the Heighway Dragon [PITH_FULL_IMAGE:figures/full_fig_p025_13.png]

**Figure 14.** Figure 14: Beginning of the first fold 25 [PITH_FULL_IMAGE:figures/full_fig_p025_14.png]

**Figure 15.** Figure 15: First fold, at 90 degrees After repeated foldings, we will open each fold to 90 degrees. If we then send the ant on another journey from A to B, it would turn 90 degrees left (counterclockwise) at C. It is not yet time to unfold, however. So far, we have made one comp…

**Figure 16.** Figure 16: First fold, completed Next, we make our second fold. To start, we insert two new vertexes, D1 and D2, at the midpoints of AC and CB respectively. (In the figures below, we will represent the “paper” as partially unfolded, so that we can see which vertex and edge is wh…

**Figure 17.** Figure 17: Beginning of second fold (with non-overlapping vertexes for clarity) [PITH_FULL_IMAGE:figures/full_fig_p026_17.png]

**Figure 18.** Figure 18: Second fold, partially complete (with non-overlapping vertexes for clarity) [PITH_FULL_IMAGE:figures/full_fig_p027_18.png]

**Figure 19.** Figure 19: After two folds, opened to 90 degrees We learn a number of things from just these first two folds. 1. A vertex inserted into an edge that the ant would traverse from left to right (were the figure in its fully-folded state) becomes a left turn when all folds are opene…

**Figure 20.** Figure 20: The Heighway Dragon after four iterations [PITH_FULL_IMAGE:figures/full_fig_p028_20.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

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