Pith. sign in

REVIEW 5 minor 17 references

A $3$-adic Recurrence for the Fixed Points of the Josephus Function $J_4$

T0 review · 0 major / 5 minor · reviewed 2026-07-12 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The fixed points of the Josephus function J4 obey a 3-adic recurrence built from alternating blocks of near-misses.

desk verdict Solid, fully proved 3-adic recurrence for the fixed points of J4; the two-type block structure and valuation meters are new and usable, with only the expected dependence on the authors’ earlier local-linearity engine. read the letter →

arxiv 2607.01270 v3 pith:TYX5NP3R submitted 2026-06-30 math.GM

Yunier Bello-Cruz , Roy Quintero-Contreras This is my paper · ORCID
classification math.GM MSC 11B3711A0711B8305A99
keywords Josephusproblemfunctionfixedpoint3-adicvaluationrecurrencehighextremalintegersequence
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

In the Josephus problem with every fourth person eliminated, certain circle sizes leave the last-seated person as the sole survivor. Those fixed-point sizes grow with erratic gaps. This paper shows that the near-misses between consecutive fixed points—sizes where the survivor lands one or two seats short—cluster into alternating blocks of the two kinds, and that each block’s length is exactly a 3-adic valuation of a linear form in the preceding circle size. Iterating those valuations produces a complete recurrence from one fixed point to the next. The same block structure also lets one evaluate the survivor for an arbitrary circle size by walking only the near-misses of a single interval, rather than the classical linear recursion through every intermediate size. Stepsize four is the first case in which two pure types coexist, and their forced alternation is what distinguishes it from the already-solved stepsizes two and three.

What carries the argument

The transition meters u3(4ne+3) (type-1 o2) and u3(4ne+5) (type-2 o1), proved by floor-by-floor divisibility diagrams on the offset sequences At and Bt and then collapsed to single valuations; together with the first-block formula and the termination rule they carry each fixed point to the next.

What would settle it

Compute the fixed-point sequence of J4 independently by the classical recurrence up to several thousand and check whether every successive gap matches the block lengths and terminal points predicted by the 3-adic valuations of Theorem 36.

Watch

Extended reading notes

Core claim

For consecutive fixed points of J4 at least 3, the pure high extremal points between them form a finite alternating sequence of type-1 and type-2 blocks whose lengths, last points, and exit to the next fixed point are completely determined by the ternarity of the starting fixed point, the valuations u3(4np+2) or u3(4np+3), the transition meters u3(4ne+3) and u3(4ne+5), and an explicit termination rule—without computing any intermediate values of J4.

Load-bearing premise

Everything rests on the local-linearity and type-value identities for high extremal points of J4, taken from the authors’ earlier general theory without re-proof.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper studies fixed points of the Josephus function J_4 (stepsize 4), i.e., n with J_4(n)=n. It proves that between consecutive fixed points n_p^(\ell) \ge 3 the pure high extremal points (near-misses with J_4(n_e)=n_e-1 or n_e-2) form a finite alternating sequence of type-1 and type-2 blocks. The type and length of the first block are fixed by the ternarity of n_p^(\ell) together with a single 3-adic valuation \nu_3(4n_p^(\ell)+2) or \nu_3(4n_p^(\ell)+3) (Proposition 19). Subsequent block lengths are given by transition meters \mu_{1\to2} and \mu_{2\to1}, which collapse to the valuations \nu_3(4n_e+3) and \nu_3(4n_e+5) (Corollary 32). A termination rule (Proposition 34) converts the last pure point into the next fixed point. The resulting structural theorem (Theorem 36) yields a recurrence that advances from one fixed point to the next without intermediate values of J_4, and as a byproduct supplies an O(number of pure points in one interval) evaluation of J_4(n) via a walk along high extremal points (Theorem 38).

Significance. If correct, the result completes the fixed-point recurrence program for the first three nontrivial stepsizes (k=2,3,4), isolating the new phenomenon that appears at k=4: coexistence of two pure types forces an alternating chain of valuations rather than a single index. The proofs are fully elementary and constructive once the local-linearity engine of Lemmas 4 and 13 is granted; every block length is an explicit 3-adic valuation of a linear form, and the evaluation algorithm of §6.3 is immediately usable. The work also generates three new OEIS sequences recording block counts and types, and cleanly frames the open case k=5. These are solid, self-contained contributions to the arithmetic of Josephus functions and related integer sequences.

minor comments (5)
  1. The dependence on the k=4 specializations of Theorem 5 of [2] (Lemmas 4 and 13) is legitimate but heavy. A short self-contained appendix restating the nine-row type table and the max-floor formula would make the present paper readable in isolation.
  2. Notation for the last points of blocks (n_e^(\ell,j)) and the auxiliary sequences (s_q) of Lemmas 28–29 is dense; a single summary table of all symbols introduced in §§2–5 would help the reader keep track.
  3. Example 31 recomputes the five-block interval of Example 12; the two presentations could be merged or cross-referenced more tightly to avoid repetition.
  4. In the concluding remarks the authors correctly flag that k=5 is the first composite modulus case; a one-sentence numerical check that the three predicted pure types appear up to 2·10^6 (already mentioned) could be moved into the introduction for motivation.
  5. Minor typographical points: “Halbeisen-Hungerbühler” appears with inconsistent umlaut encoding; the arXiv header date “8 Jul 2026” is presumably a placeholder and should be corrected on final submission.

Circularity Check

1 steps flagged · score 2.0 of 10

Legitimate self-citation of the authors' prior local-linearity engine for J_k; the new block/meter/valuation recurrence for k=4 is proved from those lemmas without tautological reduction.

  1. self citation load bearing [Section 4, Lemma 4 and Lemma 13 (and their use in every subsequent engine step)]
    "Lemma 4 (Type-value identity). For every high extremal point n_e ≥ 6, J_4(n_e + 1) = 3 - t, where t is the type of n_e. Proof. This is [2, Theorem 5(b)]. … Lemma 13 … Everything here is the k=4 case of Theorem 5 of [2]; we only rename that paper's residue variable r to ε."

    The entire block construction, successor-type congruence (Cor. 14), max-floor arithmetic, and therefore the main recurrence rest on these two lemmas. They are justified solely by citation to the authors' own prior paper [2] rather than re-derived from the defining recursion of J_4 inside the present manuscript. The dependence is real but not circular in the strong sense: the lemmas are elementary specializations that can be verified independently, and every later identity (valuations equal to meters, termination formulas) is proved line-by-line from them without further external premises.

full rationale

The derivation chain begins from the Josephus recursion and the type-value identity (Lemma 4) plus local linearity / next-extremal map (Lemma 13), both taken as the k=4 case of Theorem 5 of the authors' earlier paper [2]. Those lemmas are restated with full proofs specialized to k=4 (including the nine-row table of residues and the engine-step arithmetic), then used to construct first blocks (Prop. 19), floor-by-floor transition meters via offset sequences (A_t), (B_t) (Lemmas 28-29, Prop. 30), their collapse to single 3-adic valuations (Cor. 32), and the termination rule (Prop. 34). Theorem 36 simply assembles these proved pieces. No parameter is fitted to data and then re-predicted; no uniqueness theorem is imported to forbid alternatives; no equation reduces the target recurrence to an input by construction. The self-citations to [2,3,4] supply the base engine and the solved k=3 comparison, which is normal cumulative work and externally checkable against the defining recursion of J_4. Score 2 reflects only that minor, non-load-bearing dependence.

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

Pure discrete mathematics. No free parameters are fitted. The only non-standard background is the local analysis of high extremal points of Jk imported from the authors’ earlier work; everything else is standard arithmetic or definitions introduced for the paper.

assumptions (3)
  • domain assumption The Josephus recurrence J4(1)=1 and J4(n)=(J4(n-1)+3) mod n +1 for n≥2, together with the definition of fixed points as solutions of J4(n)=n.
    Standard definition of the problem; stated in Definition 1.
  • domain assumption Local linearity of J4 on the segments between consecutive high extremal points and the type-value identity J4(ne+1)=3-t (Lemma 4 / [2, Thm 5]).
    Imported without re-proof from the authors’ prior paper; used as the engine for every successor calculation.
  • standard math Standard properties of the 3-adic valuation ν3 and elementary modular arithmetic.
    Used throughout for block-length formulas.
invented entities (3)
  • Blocks of pure high extremal points (maximal same-type runs) and the associated last-point sequence n_e^(ℓ,j) independent evidence
    purpose: Organize the pure points between consecutive fixed points so that lengths can be read by successive valuations.
    Defined in Definition 7; the alternation and finiteness are proved from the successor-type congruence.
  • Transition meters μ1→2 and μ2→1 (later shown equal to ν3(4ne+3) and ν3(4ne+5)) independent evidence
    purpose: Give the length of the next opposite-type block from the terminal pure point of the current block.
    Defined in Definition 23; closed form proved in Corollary 32 via the offset sequences At, Bt.
  • Offset sequences (At) and (Bt) and the associated odd/even divisibility diagrams
    purpose: Provide the floor-by-floor certificate that constructs the pure points of each block and identifies the terminal index.
    Introduced in Section 5 solely as a proof device; they collapse to single valuations and are not needed for the final recurrence.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A $3$-adic Recurrence for the Fixed Points of the Josephus Function $J_4$." pith.science (2026). https://pith.science/paper/TYX5NP3R

@misc{pith2026260701270,
  author       = {Pith},
  title        = {Pith review of: A $3$-adic Recurrence for the Fixed Points of the Josephus Function $J_4$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TYX5NP3R}},
  note         = {Machine review of arXiv:2607.01270}
}
read the original abstract

In the Josephus problem with stepsize four, the participants in a circle are eliminated one by one, every fourth person leaving, until a single survivor remains. A fixed point occurs when the survivor turns out to be the person who began in the last seat. The circle sizes with this property form the sequence 1; 21; 38; 51; 122; 163; 689; 919; 2,906; and so on, whose gaps fluctuate erratically. This paper explains the fluctuation and turns it into a recurrence. Between consecutive fixed points, the circle sizes at which the survivor falls exactly one or two seats short of the last one, the near-misses, group into alternating blocks of the two kinds, and the length of every block is the number of times three divides a simple quantity built from the circle size that precedes the block. Iterating these divisibility counts carries each fixed point to the next. Stepsize four is the first case in which two kinds of near-miss coexist, and the alternation they force is what separates it from the solved cases of stepsizes two and three. As a byproduct, the survivor's position for an arbitrary circle size can be computed by walking the near-misses of a single interval, in a number of steps proportional to their count, rather than stepping through every smaller circle as the defining recursion does.

Figures

Figures reproduced from arXiv: 2607.01270 by the authors.

Figure 1
Figure 1. The Josephus function J4(n) for 1 ≤ n ≤ 165, plotted as discrete values. Each climb of slope 4 rises toward the diagonal y = n and is capped by a high extremal point. The fixed points shown are n = 1; 21; 38; 51; 122; 163; between them sit the pure high extremal points, separated by type: type-1 points (J4(n) = n − 1) at n = 6; 28; 68; 91, and type-2 points (J4(n) = n − 2) at n = 8; 11; 15. Pure points are marked on… view at source ↗
Figure 2
Figure 2. Divisibility diagram associated with the offsets ( [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 2
Figure 2. Divisibility diagram associated with the offsets ( [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figures from the paper (2 more)
Figure 3
Figure 3. Figure 3: Divisibility diagram associated with the offsets ( [PITH_FULL_IMAGE:figures/full_fig_p017_3.png]
Figure 3
Figure 3. Figure 3: Divisibility diagram associated with the offsets ( [PITH_FULL_IMAGE:figures/full_fig_p018_3.png]

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references

  1. [1]

    C. G. Bachet,Probl` emes plaisants et d´ electables qui se font par les nombres, 2nd ed., Pierre Rigaud & Associates, Lyon, 1624

  2. [2]

    Bello-Cruz and R

    Y. Bello-Cruz and R. Quintero-Contreras, Analytical study and efficient evaluation of the Josephus function,J. Integer Seq.27(2024), Article 24.3.8

    2024

  3. [3]

    Bello-Cruz and R

    Y. Bello-Cruz and R. Quintero-Contreras, On the recurrence formula for fixed points of the Josephus function,Integers24(2024), #A115

    2024

  4. [4]

    Bello-Cruz and R

    Y. Bello-Cruz and R. Quintero-Contreras, Fixed points of the Josephus function via fractional base expansions,Integers26(2026), #A54

    2026

  5. [5]

    Euler, Observationes circa novum et singulare progressionum genus,Novi Commen- tarii Academiae Scientiarum Petropolitanae20(1776), 123–139; reprinted inOpera Omnia, Ser

    L. Euler, Observationes circa novum et singulare progressionum genus,Novi Commen- tarii Academiae Scientiarum Petropolitanae20(1776), 123–139; reprinted inOpera Omnia, Ser. I,7, 246–261

  6. [6]

    R. L. Graham, D. E. Knuth, and O. Patashnik,Concrete Mathematics: A Foundation for Computer Science, 2nd ed., Addison–Wesley, Reading, MA, 1994

    1994

  7. [7]

    Halbeisen and N

    L. Halbeisen and N. Hungerb¨ uhler, The Josephus problem,J. Th´ eor. Nombres Bordeaux 9(1997), 303–318

    1997

  8. [8]

    Flavius Josephus,The Jewish War, translated by G. A. Williamson, revised with an introduction and notes by M. E. Smallwood, Penguin Classics, London, 1981

    1981

Show all 17 references
  1. [9]

    D. E. Knuth,The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed., Addison–Wesley, Reading, MA, 1997

    1997

  2. [10]

    E. L. Lloyd, AnO(nlogm) algorithm for the Josephus problem,J. Algorithms4(1983), 262–270

    1983

  3. [11]

    A. M. Odlyzko and H. S. Wilf, Functional iteration and the Josephus problem,Glasgow Math. J.33(1991), 235–240

    1991

  4. [12]

    W. J. Robinson, The Josephus problem,Math. Gaz.44(1960), 47–52

    1960

  5. [13]

    Ruskey and A

    F. Ruskey and A. Williams, The Feline Josephus problem,Theory Comput. Syst.50 (2012), 20–34

    2012

  6. [14]

    N. J. A. Sloane et al.,The On-Line Encyclopedia of Integer Sequences, published elec- tronically athttps://oeis.org, 2026

    2026

  7. [15]

    P. G. Tait, On the generalization of Josephus’ problem,Proc. Roy. Soc. Edinburgh22 (1899), 165–168

  8. [16]

    Th´ eriault, Generalizations of the Josephus problem,Util

    N. Th´ eriault, Generalizations of the Josephus problem,Util. Math.58(2000), 161–173. 33

    2000

  9. [17]

    Uchiyama, On the generalized Josephus problem,Tsukuba J

    S. Uchiyama, On the generalized Josephus problem,Tsukuba J. Math.27(2003), 319– 339. 2020Mathematics Subject Classification: Primary 11B37; Secondary 11A07, 11B83, 05A99. Keywords:Josephus problem, Josephus function, fixed point, 3-adic valuation, recurrence, high extremal poi...

    2003

Pith tools

Reviewed July 12, 2026 · model on record in the stance chip above.