pith. sign in

arxiv: 2210.02722 · v4 · submitted 2022-10-06 · 🧮 math.CO

On Frobenius Numbers of Shifted Power Sequences

Feihu Liu , Guoce Xin This is my paper

Pith reviewed 2026-05-24 11:02 UTC · model grok-4.3

classification 🧮 math.CO
keywords frobenius numbershifted square sequencespiecewise quadratic polynomialnumerical semigroupslagrange four-square theoremgenerating functionscoin problem
0
0 comments X

The pith

The Frobenius number for any shifted square sequence equals an explicit piecewise quadratic polynomial in the shift parameter a, split by residue classes modulo k squared.

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

The paper settles the open problem of finding the largest integer that cannot be written as a non-negative integer combination of the terms in A = (a, a + 1², …, a + k²). It does so by confirming a 2007 conjecture with a closed-form expression for this Frobenius number g(A). The expression is quadratic in a and changes according to the value of a modulo k². A reader would care because the general Frobenius problem is intractable, yet this family now admits an immediate, exact answer for any fixed k.

Core claim

Combining a combinatorial reduction of the membership problem to an optimization task with Lagrange's four-square theorem and generating-function methods yields an explicit formula for g(A) that is a piecewise quadratic polynomial in a, with the pieces indexed by the residue of a modulo k².

What carries the argument

Combinatorial reduction of the Frobenius problem to an optimization problem, solved via Lagrange's four-square theorem and generating-function analysis.

If this is right

  • For every fixed k the value of g(A) is obtained by direct substitution of a into one of finitely many quadratic formulas.
  • The same formula applies uniformly to all positive integers a and k without additional restrictions.
  • Each residue class modulo k² produces its own quadratic polynomial whose coefficients depend only on k and the class.
  • The conjecture of Einstein et al. holds in full generality once the piecewise expression is verified.

Where Pith is reading between the lines

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

  • The method may adapt to sequences shifted by higher even powers once an analogous optimization reduction is found.
  • The residue-class splitting points to hidden periodicity in the gaps of the numerical semigroup generated by A.
  • Independent machine verification of the formula for small k would provide a practical check before larger-scale use.

Load-bearing premise

The reduction of membership questions for these sequences to a solvable optimization problem remains valid for all a and k and introduces no hidden case restrictions when the four-square theorem and generating functions are applied.

What would settle it

For k = 3 and a = 5, compute g(A) by exhaustive search of small non-negative combinations and check whether the result equals the quadratic expression the formula assigns to the residue class 5 mod 9.

read the original abstract

We resolve the open problem of characterizing the Frobenius number $g(A)$ for shifted square sequences $A = (a, a+1^2, \ldots, a+k^2)$, confirming a conjecture of Einstein et al. (2007). By combining a combinatorial reduction to an optimization problem with Lagrange's Four-Square Theorem and generating function techniques, we derive an explicit formula for $g(A)$: a piecewise quadratic polynomial in $a$, classified by residue classes modulo $k^2$.

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.

Referee Report

1 major / 0 minor

Summary. The manuscript resolves the open problem of determining the Frobenius number g(A) for the shifted square sequence A = {a + i² | 0 ≤ i ≤ k} by deriving an explicit formula: a piecewise quadratic polynomial in a, classified according to the residue class of a modulo k². The derivation combines a combinatorial reduction of the problem to an optimization task, an application of Lagrange's Four-Square Theorem inside that optimization, and generating-function analysis; the result confirms a 2007 conjecture of Einstein et al.

Significance. If the central derivation is complete, the result supplies the first closed-form characterization of these Frobenius numbers and thereby advances the study of numerical semigroups generated by quadratic shifts. The explicit use of Lagrange's theorem and generating functions is a methodological strength that keeps the argument within standard tools of additive combinatorics.

major comments (1)
  1. [reduction and optimization section (likely §3–4)] The combinatorial reduction to an optimization problem (the load-bearing step identified in the stress-test note) must be shown to enumerate every maximal non-representable integer exactly. In particular, it is necessary to verify that every integer requiring more than four summands from A, or lying in a residue class where a four-square representation cannot be aligned with the generators of A, is still captured by the subsequent generating-function enumeration; otherwise the claimed piecewise quadratic will omit cases even while appearing correct on the examined residues mod k².

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed reading of the manuscript and for identifying a point that requires explicit clarification in the proof of the main result. We address the major comment below and will revise the manuscript to strengthen the exposition of the completeness argument.

read point-by-point responses

  1. Referee: The combinatorial reduction to an optimization problem (the load-bearing step identified in the stress-test note) must be shown to enumerate every maximal non-representable integer exactly. In particular, it is necessary to verify that every integer requiring more than four summands from A, or lying in a residue class where a four-square representation cannot be aligned with the generators of A, is still captured by the subsequent generating-function enumeration; otherwise the claimed piecewise quadratic will omit cases even while appearing correct on the examined residues mod k².

    Authors: We agree that the completeness of the reduction must be made fully explicit. The optimization step, combined with Lagrange's theorem, identifies the largest candidates in each residue class modulo k² by considering representations with at most four squares; the generating-function analysis then enumerates all possible sums from A without any a priori bound on the number of summands. This enumeration is performed directly via the coefficient extraction in the relevant generating function, which by construction accounts for arbitrary numbers of terms and for all residue classes. Consequently, any integer that cannot be represented (including those requiring more than four summands or whose four-square decompositions cannot be aligned) is detected as a non-representable element and is incorporated into the piecewise quadratic formula. To address the referee's concern, we will insert a short subsection (new §4.3) that explicitly verifies this coverage by showing that the generating-function count coincides with the optimization output on all residue classes and by providing a short argument that no larger non-representable integers exist outside the enumerated set. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses external theorems and independent reduction

full rationale

The paper's central derivation combines a combinatorial reduction of the Frobenius problem for the shifted sequence A to an optimization problem, Lagrange's Four-Square Theorem (a standard external result), and generating-function analysis to obtain the piecewise quadratic formula classified by residues mod k². No equation or step is shown to be self-definitional, no fitted parameter is relabeled as a prediction, and the 2007 conjecture being confirmed is external. The load-bearing steps rely on independently verifiable combinatorial and number-theoretic facts rather than self-citation chains or ansatzes smuggled from prior author work. This is the normal case of a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of an unshown combinatorial reduction to an optimization problem together with the applicability of two classical tools; no free parameters or new entities are mentioned.

axioms (1)
  • standard math Lagrange's Four-Square Theorem
    Cited in the abstract as part of the derivation.

pith-pipeline@v0.9.0 · 5598 in / 1128 out tokens · 34848 ms · 2026-05-24T11:02:14.628458+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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

    math.NT 2023-06 unverdicted novelty 6.0

    Explicit formulas for F(A) and g(A) are obtained for the semigroup generated by A = (a, ba + d, b²a + ((b²-1)/(b-1))d, ..., b^k a + ((b^k-1)/(b-1))d) when a ≥ k-1 - (d-1)/(b-1), with simplifications for Mersenne, Thab...

  2. A Combinatorial Approach to Frobenius Numbers of Some Special Sequences (Complete Version)

    math.CO 2023-03 unverdicted novelty 5.0

    A combinatorial reduction of the Frobenius problem to an optimization task produces explicit formulas for g(A), n(A), and s(A) on special sequences and applies MacMahon's partition analysis to count representations.

  3. The Frobenius Formula for $A=(a,ha+d,ha+b_2d,...,ha+b_kd)$

    math.CO 2023-04 unverdicted novelty 4.0

    Extends the stable property of Frobenius numbers to sequences A(a)=(a, ha+dB) yielding a congruence-class characterization of g(A(a)) mod bk for large a, plus explicit formulas for several B.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · cited by 3 Pith papers · 1 internal anchor

  1. [1]

    Brauer, On a problem of partitions , Amer

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

    work page 1942

  2. [2]

    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

  3. [3]

    Denham, Short generating functions for some semigroup algebras , Electron

    G. Denham, Short generating functions for some semigroup algebras , Electron. J. Comb. 10 (2003), #R36

    work page 2003

  4. [4]

    A. L. Dulmage and N. S. Mendelsohn, Gaps in the exponent set of primitive matrices , Illinois J. Math. 8 (1964), 642–656

    work page 1964

  5. [5]

    Einstein, D

    D. Einstein, D. Lichtblau, A. Strzebonski, and S. Wagon, Frobenius numbers by lattice point enumeration, Integers. 7 (2007), A15

    work page 2007

  6. [6]

    G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers , Oxford: Oxford University Press. (2008)

    work page 2008

  7. [7]

    H. A. Helfgott, Major arcs for Goldbach’s theorem, Ann. of Math. Studies, Princeton, to appear. See also: http://arxiv.org/abs/1305.2897v1

    work page internal anchor Pith review Pith/arXiv arXiv

  8. [8]

    Hujter, On the lowest value of the Frobenius number , Technical Report MN/31 Computer and Automation Inst., Hungarian Academy of Sciences

    M. Hujter, On the lowest value of the Frobenius number , Technical Report MN/31 Computer and Automation Inst., Hungarian Academy of Sciences. (1987)

    work page 1987

  9. [9]

    Kannan, Lattice translates of a polytope and the Frobenius problem , Combinatorica

    R. Kannan, Lattice translates of a polytope and the Frobenius problem , Combinatorica. 12(2) (1992), 161–177

    work page 1992

  10. [10]

    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

  11. [11]

    F. Liu, G. Xin, S. Ye, and J. Yin The Frobenius formula for A = (a, ha+d, ha+b2d, . . . , ha+bkd), Ramanujan J. (2024)

    work page 2024

  12. [12]

    J. L. Ram´ ırez Alfons´ ın,Complexity of the Frobenius problem, Combinatorica. 16(1) (1996), 143– 147

    work page 1996

  13. [13]

    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

  14. [14]

    Ribenboim, The Little Book of Big Primes , Springer-Verlag, (1991), 154–155

    P. Ribenboim, The Little Book of Big Primes , Springer-Verlag, (1991), 154–155

    work page 1991

  15. [15]

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

    work page 1956

  16. [16]

    J. B. Rosser, The n-th prime is greater than n log n, Proc. London Math. Soc. 45 (1939), 21–44

    work page 1939

  17. [17]

    J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers , Illinois J. Math. 6 (1962), 64–94

    work page 1962

  18. [18]

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

    work page 1977

  19. [19]

    J. J. Sylvester, On the partition of numbers , Quart. J. Pure Appl. Math. 1 (1857), 141–152

    work page

  20. [20]

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

    work page

  21. [21]

    Tripathi, The Frobenius problem for modified arithmetic progressions , J

    A. Tripathi, The Frobenius problem for modified arithmetic progressions , J. Integer Seq. 16(7) (2013), 13.7.4

    work page 2013

  22. [22]

    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. Appendix A. Tables 1 and 2 1,2School of Mathematical Sciences, Capital Normal University, Beijing 100048, PR China Email address : 1liufeihu7476@163.com & 2guoce xin@163.com 16 FEIHU LIU 1 AND GUOCE XIN 2,∗ Table 1. Discussion on A = (a, a + 12, a +...

    work page 2017