Rational dynamics of a prime-representing map

Andr\'e Carvalho

arxiv: 2605.21802 · v1 · pith:YVTM7QW5new · submitted 2026-05-20 · 🧮 math.NT · math.DS

Rational dynamics of a prime-representing map

Andr\'e Carvalho This is my paper

Pith reviewed 2026-05-22 08:04 UTC · model grok-4.3

classification 🧮 math.NT math.DS

keywords rational dynamicsorder of rationalsnatural densityarithmetic progressionsfloor functionfractional partrecurrence relationsresidue classes

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The pith

For any fixed denominator, reduced fractions of finite order under the map T have natural density one.

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

The paper studies iteration of the map T(x) equal to the integer part of x times one plus the fractional part of x, applied to rational numbers at least 2. It defines the order of a reduced fraction a over M as the smallest number of steps until an iterate lands on an integer. The central result is that these finite-order fractions comprise a set of natural density one inside the reduced fractions sharing any fixed denominator M. This density statement immediately rules out the possibility of an infinite arithmetic progression made entirely of infinite-order rationals.

Core claim

The reduced fractions a/M of exact order n are described exactly by certain residue classes of a modulo M to the power n+1. The count A(n,M) of such classes obeys a recurrence relation. For each fixed M the proportion of all reduced a/M that possess some finite order therefore tends to one, which precludes any infinite arithmetic progression consisting solely of infinite-order rationals. An explicit family realizing every prescribed order is constructed for arbitrary M, and the orders are completely classified when M equals 2.

What carries the argument

The order of the rational x = a/M, defined as the least nonnegative integer n such that the n-fold iterate of T applied to x is an integer, with this order tracked by the evolution of the numerator a modulo successively higher powers of the denominator M.

If this is right

The number A(n,M) of residue classes of exact order n satisfies a recurrence that can be used to compute the density of finite-order fractions for any fixed M.
For every denominator M and every positive integer n there exists at least one reduced fraction a/M of exact order n.
When the denominator is 2 the order of every reduced fraction a/2 is explicitly determined by the residue of a modulo a suitable power of 2.
There cannot exist an infinite arithmetic progression in which every term is a reduced rational of infinite order.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The residue-class description for each order n supplies an algorithmic way to decide the order of any given a/M by checking membership in successively larger moduli.
Because the density is one for every fixed M, the set of all infinite-order rationals (across all denominators) is expected to be sparse in the usual ordering of the rationals.
The recurrence for A(n,M) may admit a closed-form solution or generating function that makes the exact density for each M computable without enumeration.

Load-bearing premise

The iteration of T on a reduced fraction a/M remains a rational whose order is completely determined by the numerator modulo increasing powers of M, without extra constraints imposed by the floor function at intermediate steps.

What would settle it

An explicit reduced fraction a/M for which none of the iterates T to the n of a/M is ever an integer, or an arithmetic progression of such a modulo M to a high power in which every term has infinite order.

read the original abstract

We study the rational dynamics of the map $\mathcal{T}(x)=\lfloor x\rfloor(1+\{x\})$, which appears in the recursive construction of the prime-representing constant of Fridman, Garbulsky, Glecer, Grime and Florentin. For a rational number $x\geq 2$ with denominator $M$, we define its order to be the least non-negative integer $n$ such that $\mathcal{T}^n(x)$ is an integer, if such an $n$ exists, and ask whether every rational number has finite order. For each $n$, we prove that the reduced fractions $a/M$ of exact order $n$ are described by residue classes of $a$ modulo $M^{n+1}$, and give a recurrence for the number $A(n,M)$ of residue classes of exact order $n$. We then show that for each fixed denominator the fractions of finite order have natural density one among all reduced fractions with that denominator, which implies in particular that there is no infinite arithmetic progression of rational numbers of infinite order. We also give an explicit family of fractions of prescribed order for every denominator, and fully characterize the case $M=2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a residue-class description plus recurrence and density-one result for finite-order rationals under T, but the floor-function step needs checking to confirm the classes really control the orbits.

read the letter

The main thing here is that the paper describes the exact-order-n reduced fractions a/M by residue classes of a modulo M^{n+1}, gives a recurrence for the count A(n,M), and proves that finite-order fractions have natural density one for each fixed denominator. It also supplies an explicit family of prescribed-order examples and settles the M=2 case completely. These are the concrete new pieces. The recurrence in particular is useful because it turns the counting into something one can actually compute for small values, and the density statement cleanly rules out infinite arithmetic progressions of infinite-order rationals. The explicit family is straightforward to check and gives immediate examples. The potential soft spot is exactly the one in the stress-test note. The claim that the residue classes mod M^{n+1} determine the n-step iteration assumes the floor function does not introduce extra constraints when a is large. Within a fixed class the integer part floor(a/M) can still differ by multiples of M^n, which might change how the fractional-part multiplication feeds into the next steps. If the paper only shows the classes are closed under the map without proving that every element in the class has the same orbit length, then the actual number of finite-order classes could be smaller than A(n,M) summed over n, and the density might fall short of one. The abstract states that proofs exist for the description and the density, so presumably the argument handles the floor dependence, but that step is load-bearing and deserves a close read. This paper is for people already working on arithmetic dynamics or on the Fridman-Garbulsky-Glecer-Grime-Florentin constant. A reader who wants explicit counting tools or concrete examples in this narrow setting will find it worth their time. It is too specialized for a general audience. I would maybe bring it to a reading group if the group has someone interested in rational maps. I would not cite it in my own work in the next year. It shows clear engagement with the definitions and literature, so it deserves a serious referee even if the modular-invariance proof needs tightening. Recommendation: send it to peer review and ask the referee to verify that the iteration is fully determined by the stated congruences.

Referee Report

1 major / 2 minor

Summary. The manuscript examines the rational dynamics under the map T(x) = floor(x)(1 + {x}), which arises in the construction of a prime-representing constant. For rationals x = a/M ≥ 2 in lowest terms, the order is defined as the smallest nonnegative integer n such that the n-th iterate T^n(x) is an integer. The authors prove that the reduced fractions of exact order n are characterized by certain residue classes of a modulo M^{n+1}, and derive a recurrence relation for the count A(n,M) of these classes. They further establish that, for any fixed denominator M, the finite-order fractions have natural density one among all reduced fractions with that denominator. This has the consequence that there cannot exist an infinite arithmetic progression of rationals all of infinite order. Additionally, an explicit family of fractions of any given order is constructed, and the dynamics are completely described when M = 2.

Significance. Should the central claims hold, the work provides a rigorous framework for understanding the termination of orbits to integers under this map for rational starting points. The residue-class characterization and the recurrence for A(n,M) enable precise counting, while the density-one result is a strong statement about the prevalence of finite-order points and rules out certain infinite structures in the rationals. The explicit constructions and the complete analysis for M=2 offer concrete illustrations. These results appear to be derived from direct combinatorial arguments on the map's action rather than empirical fitting.

major comments (1)

[Residue class description and proof of density result] The assertion that fractions of exact order n correspond exactly to residue classes a modulo M^{n+1} assumes that the n-step iteration of T is completely determined by this congruence. However, since T involves the floor function, for a and a + k M^{n+1} with large k, floor((a + k M^{n+1})/M) differs by multiples of M^n. This may alter subsequent integer parts and fractional-part multiplications in the iterates in ways not encoded by the modulus alone. The proof must explicitly verify that the order is invariant under these shifts and that no extra constraints arise from the floor function (see the definition of order and the residue class description). If such constraints exist, the count of exact-order-n classes is strictly smaller than A(n,M), so the claimed natural density one need not hold.

minor comments (2)

The abstract and introduction could more explicitly recall the definition of the fractional part {x} and floor function for readers less familiar with the notation.
It would be helpful to include a small table or example computing A(n,M) for small M and n to illustrate the recurrence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting a potential issue in the proof of the residue class description. We address this comment in detail below and have revised the manuscript accordingly to improve clarity.

read point-by-point responses

Referee: The assertion that fractions of exact order n correspond exactly to residue classes a modulo M^{n+1} assumes that the n-step iteration of T is completely determined by this congruence. However, since T involves the floor function, for a and a + k M^{n+1} with large k, floor((a + k M^{n+1})/M) differs by multiples of M^n. This may alter subsequent integer parts and fractional-part multiplications in the iterates in ways not encoded by the modulus alone. The proof must explicitly verify that the order is invariant under these shifts and that no extra constraints arise from the floor function (see the definition of order and the residue class description). If such constraints exist, the count of exact-order-n classes is strictly smaller than A(n,M), so the claimed natural density one need not hold.

Authors: We are grateful for this comment, which prompted us to make the inductive step more explicit. Let x = a/M and y = b/M with a ≡ b mod M^{n+1}, so a = b + k M^{n+1} and x = y + k M^n. Then floor(x) = floor(y) + k M^n and the fractional part {x} = {y}. It follows that T(x) = floor(x)(1 + {x}) = [floor(y) + k M^n](1 + {y}) = T(y) + k M^n (1 + {y}). Since 1 + {y} = (M + (b mod M))/M, we have T(x) = T(y) + k M^{n-1} · s for some integer s. Thus, T(x) ≡ T(y) mod M^{n-1}. Continuing inductively, the j-th iterate satisfies a congruence modulo M^{n+1-j}. In particular, after n steps, the congruence is modulo M^1, which is sufficient to determine whether the result is an integer. Therefore, T^n(x) is integer if and only if T^n(y) is, so the order is invariant under the shift. This shows there are no additional constraints from the floor function, and A(n,M) correctly counts the classes. The density-one result is unaffected. We have inserted an expanded explanation of this calculation into the proof of the main theorem. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses direct counting from map definition

full rationale

The paper defines the order of a rational via the least n such that T^n(x) is integer, then proves that exact-order-n fractions a/M correspond to specific residue classes of a modulo M^{n+1} by analyzing the action of T on the numerator and denominator. It derives a recurrence for the count A(n,M) of such classes directly from the floor-and-fractional-part definition of T, without fitting parameters or renaming inputs as outputs. The density-one result for finite-order fractions follows by summing the recurrence over n and comparing to the total count of reduced fractions with denominator M. No self-citations, ansatzes, or uniqueness theorems are invoked as load-bearing steps; the chain is self-contained and externally verifiable from the map definition alone.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard properties of the floor and fractional-part functions together with basic facts about rational arithmetic and modular residue classes. No free parameters or new entities are introduced.

axioms (1)

standard math Floor and fractional-part functions satisfy their standard identities on the reals, and iteration of T preserves rationality when started from a rational.
Invoked in the definition of T and the order for rational inputs.

pith-pipeline@v0.9.0 · 5737 in / 1398 out tokens · 53656 ms · 2026-05-22T08:04:29.743895+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

Azevedo, M

A. Azevedo, M. Carvalho, and A. Machiavelo,Dynamics of a quasi-quadratic map, Jour- nal of Difference Equations and Applications20(2014), no. 1, 1–16

work page 2014
[2]

Fridman, J

D. Fridman, J. Garbulsky, B. Glecer, J. Grime, and M. Tron Florentin,A prime- representing constant, American Mathematical Monthly121(2014), no. 1, 1–4

work page 2014
[3]

Moreira,https://mathoverflow.net/questions/377706/ an-alternative-to-continued-fraction-and-applications#comment958452_ 377706 10

J. Moreira,https://mathoverflow.net/questions/377706/ an-alternative-to-continued-fraction-and-applications#comment958452_ 377706 10

work page

[1] [1]

Azevedo, M

A. Azevedo, M. Carvalho, and A. Machiavelo,Dynamics of a quasi-quadratic map, Jour- nal of Difference Equations and Applications20(2014), no. 1, 1–16

work page 2014

[2] [2]

Fridman, J

D. Fridman, J. Garbulsky, B. Glecer, J. Grime, and M. Tron Florentin,A prime- representing constant, American Mathematical Monthly121(2014), no. 1, 1–4

work page 2014

[3] [3]

Moreira,https://mathoverflow.net/questions/377706/ an-alternative-to-continued-fraction-and-applications#comment958452_ 377706 10

J. Moreira,https://mathoverflow.net/questions/377706/ an-alternative-to-continued-fraction-and-applications#comment958452_ 377706 10

work page