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The Diophantine Frobenius Problem revisited

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper proves that the gcd-constrained Frobenius number is finite for every coprime denomination set and pins the two-coin value between a1a2 and a constant times a1a2(log a1a2)^2.

desk verdict Frobenius variant with gcd-constrained variables: solid new results, one repairable proof gap in the finiteness theorem. read the letter →

arxiv 2509.08599 v1 pith:V4I4H764 submitted 2025-09-10 math.NT math.CO

Yuchen Ding , Weijia Wang , Hao Zhang This is my paper
classification math.NTmath.CO MSC 11D0711D04
keywords DiophantineFrobeniusproblemgcd-restrictedrepresentationsprimitivenumberJacobsthalfunctionMöbiusinversionnumericalsemigroupsdenumerant
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 classical Frobenius coin problem asks when all large integers are nonnegative integer combinations of coprime coin values. This paper adds the condition that the multipliers in the combination have no common divisor, and proves that the threshold G still exists for any finite set of coins. For two coins, it shows the threshold is always at least a1a2 and at most an absolute constant times a1a2(log a1a2)^2, so the price of requiring coprime multipliers is at most a polylogarithmic factor. It also gives an exact formula for the number of coprime representations of n, proves the exact value G_{2,a}=4a-2, and shows conditionally under the generalized Riemann hypothesis that the error in the formula averages as O(a1a2 N^{1/2+epsilon}). The upshot is that the Diophantine Frobenius problem remains meaningful and quantitatively close to the classical one when representations are required to be primitive.

What carries the argument

Two identities carry the proof. First, Möbius inversion turns the unrestricted representation count g into the coprime-multiplier count f: f(n)=sum_{d|n} mu(d) g(n/d). Second, for two variables the unrestricted count is explicit: g(n)=floor((n-a2 r_n)/(a1a2))+1, where r_n is determined modulo a1. To bound G, all solutions are parametrized as (x0 - k a2, y0 + k a1); a solution is non-coprime only when k lies in certain residue classes modulo primes dividing n. The Jacobsthal function, defined as the least run length of consecutive integers guaranteed to contain a number avoiding all those residue classes, then ensures a good k appears every j(n) steps, and the known bound j(n) << (log n)^2 yi

What would settle it

The two-coin upper bound would be disproved by a sequence of coprime pairs (a1,a2) with G_{a1,a2}/(a1a2(log a1a2)^2) unbounded, since the implied constant in Theorem 1.6 is absolute; such pairs could be sought by exact enumeration for increasing a2. The k-variable finiteness claim would be disproved by any finite denomination set and any infinite sequence of n for which every representation has gcd(x1,...,xk)>1; checking this for a small fixed triple up to large n is a finite computation.

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Extended reading notes

Core claim

The central claim is that the restricted representation count f(n), counting representations of n with no common divisor among the multipliers, is eventually positive. The proof obtains f(n) from the unrestricted count by Möbius inversion, f(n)=sum_{d|n} mu(d) g(n/d), and shows the main term dominates the divisor-sum error. For two coprime denominations a1<a2, the restricted count is exactly phi(n)/(a1a2)+E(n) with |E(n)|<2^{omega(n)}, and an explicit expression for E(n) is given. Consequently G_{a1,a2} is finite and satisfies a1a2 <= G_{a1,a2} << a1a2(log a1a2)^2, with an absolute implied constant. The paper also proves the exact value G_{2,a}=4a-2 for odd a>2, and constructs pairs showing

Load-bearing premise

In Section 2, the proof of Theorem 1.3 assumes the unrestricted representation count is exactly one polynomial in n with coefficients independent of n; for fixed coin values it is normally a quasi-polynomial whose coefficients repeat periodically with n, so that exact step needs replacing by the asymptotic main term c n^{k-1}+O(n^{k-2}).

Editorial extensions

If this is right

  • The finiteness theorem makes G a well-defined invariant, the 'primitive Frobenius number', for any finite set of coprime denominations, so the open question of a closed form is coherent.
  • For two denominations, the threshold is always within a polylogarithmic factor of a1a2, so any search for a coprime representation can be stopped at O(a1a2(log a1a2)^2).
  • The exact formula for f(n) lets one test whether a given n has a coprime representation by evaluating a divisor sum, with phi(n)/(a1a2) as the main approximation.
  • The exact value G_{2,a}=4a-2 supplies an infinite family of even threshold values and shows that the problem's behavior changes sharply at a1=2.
  • Conditionally on the generalized Riemann hypothesis, the error term averages as O(a1a2 N^{1/2+epsilon}), so the main-term formula is accurate on average over n.

Reading between the lines

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

  • Editorial inference: The k-variable finiteness argument uses an exact polynomial premise for the unrestricted representation count that is not literally true for fixed coin values; replacing it with the standard asymptotic c n^{k-1}+O(n^{k-2}) should preserve the positivity step, so the finiteness conclusion is likely repairable.
  • Editorial inference: The explicit f(n) formula yields a finite, though slow, algorithm to compute G_{a1,a2} exactly: scan n upward until the formula is positive for all n up to the O(a1a2(log a1a2)^2) stopping point.
  • Editorial inference: The parity data, mostly even G values with a short list of odd exceptions, suggests a modular-period structure; testing whether the odd values concentrate in particular residue classes of (a1,a2) modulo small moduli is a direct next step.
  • Editorial inference: The same Möbius-inversion technique could be applied to other restricted representability questions, such as requiring each multiplier to be positive or squarefree, though the positivity of the main term would need to be rechecked case by case.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper introduces a Diophantine Frobenius problem with the additional coprimality condition gcd(x_1,...,x_k)=1. It claims Theorem 1.3: G_{a_1,...,a_k} is finite for every coprime tuple; for k=2, Theorem 1.6 gives a_1a_2 <= G_{a_1,a_2} << a_1a_2(log a_1a_2)^2; Theorem 1.4 gives an explicit two-variable denumerant with coprimality; Theorem 1.5 gives a conditional GRH mean-value estimate for the error term; Theorem 1.7 gives a liminf lower bound; Theorem 1.8 gives G_{2,a}=4a-2. The introduction motivation is primes in numerical semigroups; the later sections also study prime powers represented by a_1x+a_2y.

Significance. If established, the paper introduces a natural finite invariant extending the classical Frobenius number and provides essentially the correct order of magnitude in the two-variable case. The proofs are mostly elementary and self-contained, with the analytic estimate in Theorem 1.5 conditional on GRH. The paper is honest in separating conditional/unconditional results and in leaving open problems. However, the proof of the central finiteness theorem relies on a false polynomial-denumerant premise; the gap is local and repairable, but the manuscript as written is not rigorous at that point.

major comments (2)
  1. [Section 2, Proof of Theorem 1.3, Eq. (2.2)] The proof asserts from [2, Eq. (1.3)] that g_{a_1,...,a_k}(n) is a single polynomial c_0+...+c_{k-1} n^{k-1} with coefficients independent of n. For fixed a_i the denumerant is only a quasi-polynomial; e.g. for a_1=2, a_2=3 the representation count is not one polynomial in n. Thus Eq. (2.2) is not justified as written. This is load-bearing, since it is used to prove G_{a_1,...,a_k} is finite. The repair is standard: replace the false premise by the asymptotic g_{a_1,...,a_k}(n)=c_{k-1}n^{k-1}+O(n^{k-2}), c_{k-1}=1/((k-1)!\prod a_i)>0. Then Möbius inversion gives f(n)=c_{k-1}n^{k-1}\prod_{p|n}(1-p^{-(k-1)})+O(n^{k-2}2^{\omega(n)}), which is positive for all large n for k>=3; for k=2 the main term is c_1\varphi(n), and \varphi(n)/2^{\omega(n)}\to\infty. The theorem is therefore likely correct, but the current proof must be corrected.
  2. [Section 3, Proof of Theorem 3.1, Eq. (3.2)] In the decomposition (3.1)-(3.2), the k=q term is written as the constant 1. For a general periodic h with |h|<=A, the k=q contribution is c_{q,1} \sum_{n<=N_1} a_q(n) = h(q) (since a_q(n)=1 only at n=q), not 1. The displayed +1 is correct for the particular h in Theorem 1.5, where h(q)=1, but the stated general Theorem 3.1 needs either a factor h(q) or an explicit normalization. This is a local proof error and does not affect the unconditional results, but it should be fixed before publication.
minor comments (4)
  1. [Theorem 1.4, displayed formula for E(n)] In the displayed error term, the numerator in the fractional part appears to be misprinted: it should be d - a_2 r_d (with the divisor index d), not n d or n/d, after changing variables in the Möbius sum. Please check and correct.
  2. [Section 4, Lemma 4.2] Typo: 'abosulte' should be 'absolute'.
  3. [Section 1, after Theorem 1.5] Typo: 'rountine' should be 'routine'; also the sentence 'from explicit bounds ... with rountine computations' could be expanded for clarity.
  4. [Proof of Theorem 1.8] The first sentence would be clearer if it explicitly said both listed representations have gcd(x_1,x_2)>1, and that these are the only possible representations of 4a-2.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proofs are self-contained modulo external lemmas; the only substantive issue is a repairable non-circular correctness gap in the denumerant premise.

full rationale

The derivation chain is not circular. Theorem 1.3 is proved from the Möbius inversion identity (2.1), an external denumerant formula quoted from Bell [2], and positivity of the resulting main term for k>=3 or phi(n) for k=2; none of these inputs is the desired finiteness conclusion. The two-variable results are derived from the explicit parametrization in (4.1), Lemma 2.1, Iwaniec's Jacobsthal bound, CRT, PNT, and standard Perron/L-function estimates under GRH; they do not presuppose the target bounds on G. The self-citations [6], [7], [8], and [9] appear only as introductory literature context and are not load-bearing in any proof. The reader-identified flaw is real but is a correctness gap, not circularity: Bell's Eq. (1.3), as quoted in the proof of Theorem 1.3, treats the denumerant as a single polynomial, while for fixed coin values it is only a quasi-polynomial. This is repairable by replacing the false polynomial premise with the standard asymptotic c_{k-1} n^{k-1} + O(n^{k-2}), which still yields f(n)>0 for large n. Because no fitted parameter is renamed as a prediction and no self-citation is used to force a conclusion, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard theorems and an explicitly stated GRH assumption. The one nonstandard premise is the polynomial-denumerant assertion, which is misstated but replaceable by a quasi-polynomial asymptotic; all other external results are standard and properly cited.

assumptions (4)
  • ad hoc to paper The denumerant g_{a1..ak}(n) is a polynomial in n of degree k-1 with coefficients independent of n.
    Used in proof of Theorem 1.3 (Section 2). For fixed a_i the representation count is a quasi-polynomial, not a single polynomial; for k=2 Lemma 2.1 gives an explicit quasi-polynomial. The proof should instead use g(n)=c_{k-1} n^{k-1}+O(n^{k-2}).
  • domain assumption Generalized Riemann hypothesis (GRH).
    Assumed in Theorem 1.5 and Theorem 3.1 to control L(s,chi) and 1/ζ(s) on the line Re(s)=1/2+epsilon. Stated explicitly in the paper.
  • standard math Iwaniec's Jacobsthal bound j(n) << (log n)^2.
    Cited as Lemma 4.2 and used in the upper bound for G_{a1,a2}.
  • standard math Dirichlet's theorem on primes in arithmetic progressions and the prime number theorem for θ(x)~x.
    Used in Theorem 1.7 to produce infinitely many primes q not congruent to -1 mod a1 and to bound the product of primes up to a1 q.

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Pith. "Pith review of The Diophantine Frobenius Problem revisited." pith.science (2026). https://pith.science/paper/V4I4H764

@misc{pith2026250908599,
  author       = {Pith},
  title        = {Pith review of: The Diophantine Frobenius Problem revisited},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V4I4H764}},
  note         = {Machine review of arXiv:2509.08599}
}
abstract

Let $k\ge 2$ and $a_1, a_2, \cdots, a_k$ be positive integers with \[ \gcd(a_1, a_2, \cdots, a_k)=1. \] It is proved that there exists a positive integer $G_{a_1, a_2, \cdots, a_k}$ such that every integer $n$ strictly greater than it can be represented as the form \[ n=a_1x_1+a_2x_2+\cdots+a_kx_k, \quad (x_1, x_2, \cdots, x_k\in\mathbb{Z}_{\ge 0},~\gcd(x_1, x_2, \cdots, x_k)=1). \] We then investigate the size of $G_{a_1, a_2}$ explicitly. Our result strengthens the primality requirement of $x$'s in the classical Diophantine Frobenius Problem.

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