On the Frobenius Number and Genus of a Collection of Semigroups Generalizing Repunit Numerical Semigroups

Feihu Liu; Guoce Xin; Jingjing Yin; Suting Ye

arxiv: 2306.03459 · v4 · submitted 2023-06-06 · 🧮 math.NT · math.CO

On the Frobenius Number and Genus of a Collection of Semigroups Generalizing Repunit Numerical Semigroups

Feihu Liu , Guoce Xin , Suting Ye , Jingjing Yin This is my paper

Pith reviewed 2026-05-24 08:52 UTC · model grok-4.3

classification 🧮 math.NT math.CO

keywords numerical semigroupsFrobenius numbergenusrepunit semigroupsMersenne semigroupsThabit semigroupsProth semigroupscoprime generators

0 comments

The pith

For a parametrized family of generators generalizing repunits, closed formulas for the Frobenius number and genus hold once the first generator meets a lower bound involving the other parameters.

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

The paper considers numerical semigroups generated by sequences A of the specific form (a, ba + d, b²a + ((b²-1)/(b-1))d, … , b^k a + ((b^k-1)/(b-1))d) where d may be negative. It derives explicit expressions for both the Frobenius number F(A), the largest integer outside the semigroup, and the genus g(A), the count of positive integers outside it, provided the generators are coprime and a satisfies a ≥ k-1 - (d-1)/(b-1). These expressions recover earlier results on repunit, Mersenne, and Thabit semigroups and give a partial answer to an open question on Proth semigroups.

Core claim

When A takes the stated geometric form in the parameters a, b, d, and k, the Frobenius number F(A) and genus g(A) admit closed formulas whenever the generators are relatively prime and the inequality a ≥ k-1 - (d-1)/(b-1) is satisfied.

What carries the argument

The specific generator sequence A = (a, ba + d, b²a + ((b²-1)/(b-1))d, … , b^k a + ((b^k-1)/(b-1))d), whose linear structure in powers of b permits direct counting of semigroup gaps.

If this is right

The formulas give immediate values of F(A) and g(A) for every member of the family that satisfies the bound.
Specialization to d=1 recovers and extends results on repunit semigroups.
The same formulas cover the Mersenne and Thabit cases as direct substitutions.
A portion of the open problem on Proth numerical semigroups receives an explicit solution.

Where Pith is reading between the lines

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

The same parametrization technique could be tested on generators obeying other linear recurrences beyond powers of a fixed base b.
Allowing negative d suggests that similar closed forms might exist for semigroups whose generators straddle multiples of a modulus in both directions.
The bound on a may be sharp; testing the boundary case a exactly equal to the threshold could reveal whether the formulas remain valid or require adjustment.

Load-bearing premise

The generators must remain relatively prime and the first generator a must meet or exceed the bound k-1 minus (d-1) divided by (b-1).

What would settle it

Pick concrete integers a, b>1, d, k>1 satisfying the inequality and gcd condition, compute the actual Frobenius number by enumeration of the semigroup, and check whether it equals the paper's claimed formula.

read the original abstract

Let $A=(a_1, a_2, \ldots, a_n)$ be a sequence of relative prime positive integers with $a_i\geq 2$. The Frobenius number $F(A)$ is the largest integer not belonging to the numerical semigroup $\langle A\rangle$ generated by $A$. The genus $g(A)$ is the number of positive integer elements not in $\langle A\rangle$. The Frobenius problem is to determine $F(A)$ and $g(A)$ for a given sequence $A$. In this paper, we study the Frobenius problem of $A=\left(a,h_1a+b_1d,h_2a+b_2d,\ldots,h_ka+b_kd\right)$ with some restrictions. An innovation is that $d$ can be a negative integer. In particular, when $A=\left(a,ba+d,b^2a+\frac{b^2-1}{b-1}d,\ldots,b^ka+\frac{b^k-1}{b-1}d\right)$, we obtain formulas for $F(A)$ and $g(A)$ when $a\geq k-1-\frac{d-1}{b-1}$. Our formulas simplify further for some special cases, such as Mersenne, Thabit, and repunit numerical semigroups. Finally, we partially solve an open problem for the Proth numerical semigroup.

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

This paper derives explicit closed forms for the Frobenius number and genus of a specific arithmetic-geometric generator family that allows negative d, under a stated lower bound on a.

read the letter

The main takeaway is that the authors produce formulas for F(A) and g(A) when the generators follow the pattern A = (a, ba + d, b²a + ((b²-1)/(b-1))d, …) with d allowed to be negative, provided a meets or exceeds k-1 - (d-1)/(b-1) and the set remains coprime. That extension to negative d and the general derivation appear to be the actual addition beyond the repunit literature they cite. The specializations to Mersenne, Thabit, and repunit cases follow immediately, and the partial result on the Proth semigroup is a modest extra step. The work stays inside the regime where the formulas are asserted to hold, so there is no overclaim visible from the abstract and stress-test note. The central algebraic reductions seem to rest on the generator structure itself rather than fitted parameters, which keeps the circularity burden low. The size condition is restrictive for small a or certain d, but the paper does not pretend the formulas apply outside it. Coprimeness still needs separate verification in applications, yet that is standard for this subfield. Overall the derivations look like routine but careful manipulation once the form is fixed; the value is in recording the general case explicitly. This is niche material aimed at people who already work on closed-form Frobenius problems for parameterized numerical semigroups. A reader hunting for new solvable families will get concrete expressions they can use or extend. It is not foundational, but it is honest incremental progress that fills a small gap. The paper deserves a serious referee to verify the algebra and check the Proth claim against the open problem statement. I would send it out rather than desk-reject.

Referee Report

1 major / 0 minor

Summary. The paper claims to study the Frobenius problem for sequences A=(a, h1 a + b1 d, …, hk a + bk d) with restrictions on the coefficients, allowing d to be negative. In particular, for the specific form A=(a, ba+d, b²a+((b²-1)/(b-1))d, …, b^k a + ((b^k-1)/(b-1))d) it obtains explicit formulas for the Frobenius number F(A) and genus g(A) whenever the generators are relatively prime and a ≥ k-1 - (d-1)/(b-1). The formulas are shown to simplify in special cases (Mersenne, Thabit, repunit) and the work partially resolves an open question on the Proth numerical semigroup.

Significance. If the claimed closed forms are rigorously derived and hold under the stated restrictions, the results would constitute a concrete advance in the Frobenius problem by supplying explicit, computable expressions for an infinite parameterized family that properly generalizes repunit semigroups. The extension to negative d and the partial resolution of the Proth case are additional points of interest.

major comments (1)

[Abstract] Abstract and main claims: the manuscript asserts that formulas for F(A) and g(A) are obtained under the listed restrictions, yet the provided text contains no derivation steps, intermediate lemmas, error bounds, or concrete verification examples that would allow independent checking of the algebraic manipulations. This absence is load-bearing for the central claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need for greater transparency in the derivations. We address the single major comment below and will incorporate the requested clarifications in a revised version.

read point-by-point responses

Referee: [Abstract] Abstract and main claims: the manuscript asserts that formulas for F(A) and g(A) are obtained under the listed restrictions, yet the provided text contains no derivation steps, intermediate lemmas, error bounds, or concrete verification examples that would allow independent checking of the algebraic manipulations. This absence is load-bearing for the central claim.

Authors: We agree that the current manuscript version presents the closed-form expressions for F(A) and g(A) with insufficient intermediate steps visible to the reader. While the proofs exist in outline form within the body, they lack the expanded lemmas, explicit algebraic manipulations, error bounds, and numerical verification examples needed for independent checking. We will revise the manuscript by inserting a dedicated section with these details, including step-by-step derivations of the formulas under the stated condition a ≥ k-1 - (d-1)/(b-1), together with concrete examples for the Mersenne, Thabit, and repunit cases. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives explicit formulas for F(A) and g(A) directly from the arithmetic structure of the given generators A = (a, ba + d, b²a + ((b²-1)/(b-1))d, …) under the stated inequality a ≥ k-1 - (d-1)/(b-1) and coprimality. No step reduces a claimed prediction to a fitted parameter by construction, no load-bearing uniqueness theorem is imported via self-citation, and the boundary condition is presented as an explicit restriction rather than derived from the result itself. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard definition of numerical semigroups generated by relatively prime integers; no new free parameters are introduced beyond the input sequence A, and no new entities are postulated.

axioms (1)

domain assumption A consists of relatively prime positive integers each at least 2, so that the generated semigroup is cofinite.
Invoked in the opening paragraph to guarantee that F(A) and g(A) are finite.

pith-pipeline@v0.9.0 · 5801 in / 1466 out tokens · 26693 ms · 2026-05-24T08:52:02.323817+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages · 1 internal anchor

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Tripathi, On the Frobenius problem for {a, ha+ d, ha+ bd, ha+ b2d,

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S. Ugolini, On numerical semigroups closed with respect to the action of affine maps, Publ. Math. Debrecen. 90 (2017), 149–167. 1,2,3,4School of Mathematical Sciences, Capital Normal University, Beijing 100048, PR China Email address : 1liufeihu7476@163.com & 2guoce xin@163.com & 3yesuting0203@163.com & 4yinjingj@163.com

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[1] [1]

Adamaszek and M

A. Adamaszek and M. Adamaszek, Combinatorics of the change-making problem , European. J. Comb. 31 (2010), 47–63

work page 2010

[2] [2]

A. Assi, M. D’Anna, and P. A. Garc´ ıa-S´ anchez,Numerical Semigroups and Applications, Second Edition. Vol.3, RSMS Springer, Cham (2020)

work page 2020

[3] [3]

M. B. Branco, I. Colaco, and I. Ojeda, The Frobenius problem for generalized repunit numerical semigroups, Mediterr. J. Math. 20 (2023), Article 16

work page 2023

[4] [4]

Brauer, On a problem of partitions , Amer

A. Brauer, On a problem of partitions , Amer. J. Math. 64 (1942), 299–312

work page 1942

[5] [5]

Brauer and J

A. Brauer and J. E. Shockley, On a problem of Frobenius , J. Reine Angew. Math. 211 (1962), 215–220

work page 1962

[6] [6]

L. J. Cowen, R. Cowen, and A. Steinberg, Totally greedy coin sets and greedy obstructions , Electron. J. Comb. 15 (2008), #R90

work page 2008

[7] [7]

Curtis, On formulas for the Frobenius number of a numerical semigroup , Math

F. Curtis, On formulas for the Frobenius number of a numerical semigroup , Math. Scand. 67 (1990), 190–192. 22 FEIHU LIU 1, GUOCE XIN 2,∗, SUTING YE 3 AND JINGJING YIN 4

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numericalsgps

M. Delgado, P. A. Garc´ ıa-S´ anchez, and J. Morais, “numericalsgps”: a gap package on numerical semigroups , Version 1.2.0 dev (2019), (Refereed GAP package), http://gap- packages.github.io/numericalsgps

work page 2019

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Gu and X

Z. Gu and X. Tang, The Frobenius problem for a class of numerical semigroups , Int. J. Number Theory. 13 (2017), 1335–1347

work page 2017

[10] [10]

Gu, On the numerical semigroup generated by {bn+1+i + bn+i−1 b−1 | i ∈ N}, Discrete Math

Z. Gu, On the numerical semigroup generated by {bn+1+i + bn+i−1 b−1 | i ∈ N}, Discrete Math. Appl. 30(4) (2020), 257–264

work page 2020

[11] [11]

T. C. Hu and M. L. Lenard, Optimality of a Heuristic solution for a class of Knapsack problems , Operations Research. 24(1) (1976), 193–196

work page 1976

[12] [12]

Lepilov, J

M. Lepilov, J. O’Rourke, and I. Swanson, Frobenius numbers of numerical semigroups generated by three consecutive squares or cubes , Semigroup Forum. 91 (2015), 238–259

work page 2015

[13] [13]

On Frobenius Numbers of Shifted Power Sequences

F. Liu and G. Xin, On Frobenius formulas of power sequences , arXiv:2210.02722, (2022)

work page internal anchor Pith review Pith/arXiv arXiv 2022

[14] [14]

Liu and G

F. Liu and G. Xin, A combinatorial approach to Frobenius numbers of some special sequences , Adv. Appl. Math. 158 (2024), 102719

work page 2024

[15] [15]

F. Liu, G. Xin, S. Ye, and J. Yin, A note on generalized repunit numerical semigroups , arXiv:2306.10738. (2023)

work page arXiv 2023

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M. J. Magazine, G. L. Nemhauser, and L. E. Trotter Jr, When the greedy solution solves a class of Knapsack problems , Operations Research. 23(2) (1975), 207–217

work page 1975

[17] [17]

J. M. Mar´ ın, J. L. Ram´ ırez Alfons´ ın, and M. P. Revuelta,On the Frobenius number of Fibonacci numerical semigroups, Integers. 7 (2007), A14

work page 2007

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D. C. Ong and V. Ponomarenko, The Frobenius number of geometric sequences , Integers. 8 (2008), A33

work page 2008

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J. L. Ram´ ırez Alfons´ ın,The Diophantine Frobenius Problem , Oxford Lecture Series in Mathe- matics and Its Applications, vol. 30, Oxford University Press. (2005)

work page 2005

[20] [20]

J. B. Roberts, Note on linear forms , Proc. Amer. Math. Soc. 7 (1956), 465–469

work page 1956

[21] [21]

A. M. Robles-P´ erez and J. C. Rosales, The Frobenius number for sequences of triangular and tetrahedral numbers, J. Number Theory. 186 (2018), 473–492

work page 2018

[22] [22]

¨O. J. R¨ odseth,On a linear diophantine problem of Frobenius II , J. Reine Angew Math. 307/308 (1979), 431–440

work page 1979

[23] [23]

J. C. Rosales and M. B. Branco, Numerical semigroups that can be expressed as an intersection of symmetric numerical semigroups , J. Pure Appl. Algebra. 171 (2002), 303–314

work page 2002

[24] [24]

J. C. Rosales, M. B. Branco, and D. Torr˜ ao, The Frobenius problem for Thabit numerical semi- groups, J. Number Theory. 155 (2015), 85–99

work page 2015

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J. C. Rosales, M. B. Branco, and D. Torr˜ ao,The Frobenius problem for repunit numerical semi- groups, Ramanujan J. 40 (2016), 323–334

work page 2016

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J. C. Rosales, M. B. Branco, and D. Torr˜ ao, The Frobenius problem for Mersenne numerical semigroups, Math. Z. 286 (2017), 1–9

work page 2017

[27] [27]

J. C. Rosales and P. A. Garc´ ıa-S´ anchez,Numerical Semigroups, Developments in Mathematics. Vol.20, Springer, New York (2009)

work page 2009

[28] [28]

E. S. Selmer, On the linear Diophantine problem of Frobenius , J. Reine Angew. Math. 293/294 (1977), 1–17

work page 1977

[29] [29]

Korean Math

Kyunghwan Song, The Frobenius problem for numerical semigroups generated by the Thabit numbers of the first, second kind base b and the Cunningham numbers , Bull. Korean Math. Soc. 57 (2020), 623–647

work page 2020

[30] [30]

21 (2021), A17

Kyunghwan Song, The Frobenius problem for extended Thabit numerical semigroups , Integers. 21 (2021), A17

work page 2021

[31] [31]

Srivastava and D

P. Srivastava and D. Thakkar, The Frobenius problem for the Proth numbers , Algorithms and Discrete Applied Mathematics. Lecture Notes in Computer Science, Vol 14508. Springer, Cham. (2024). FROBENIUS NUMBER AND GENUS 23

work page 2024

[32] [32]

J. J. Sylvester, On sub-invariants, i.e., semi-invariants to binary quantities of an unlimited order , Amer. J. Math. 5 (1882), 119–136

work page

[33] [33]

Tripathi, On the Frobenius problem for {a, ha+ d, ha+ bd, ha+ b2d,

A. Tripathi, On the Frobenius problem for {a, ha+ d, ha+ bd, ha+ b2d, . . . , ha+ bkd}, J. Number Theory. 162 (2016), 212–223

work page 2016

[34] [34]

Tripathi, Formulate for the Frobenius number in three variables , J

A. Tripathi, Formulate for the Frobenius number in three variables , J. Number Theory. 170 (2017), 368–389

work page 2017

[35] [35]

Ugolini, On numerical semigroups closed with respect to the action of affine maps, Publ

S. Ugolini, On numerical semigroups closed with respect to the action of affine maps, Publ. Math. Debrecen. 90 (2017), 149–167. 1,2,3,4School of Mathematical Sciences, Capital Normal University, Beijing 100048, PR China Email address : 1liufeihu7476@163.com & 2guoce xin@163.com & 3yesuting0203@163.com & 4yinjingj@163.com

work page 2017