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On the Frobenius Number of Quotients of Numerical Semigroups

This paper resolves a long-standing open problem regarding the Frobenius number of quotients of numerical semigroups by proving that no uniform polynomial or rational formula exists for $g(\langle a,b\rangle/p)$ , while demonstrating that for fixed $p$ , the function becomes a quadratic quasi-polynomial and satisfies no nontrivial polynomial relation when $b=a+1$ .

Original authors: Feihu Liu

Published 2026-07-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Feihu Liu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Great Number Hunt: Why Some Patterns Refuse to Be Tamed

Imagine you are a chef trying to make a specific number of cookies using only two sizes of cookie cutters, say 7-inch and 11-inch circles. You can stack them, layer them, or combine them in any way you like, but you can't cut them into smaller pieces. The "Frobenius number" is the biggest number of cookies you simply cannot make with those two cutters. For example, if you have 3-inch and 5-inch cutters, you can make 3, 5, 6, 8, 9, 10, and so on, but you can't make 7. So, 7 is your Frobenius number. Mathematicians have known for over a century how to calculate this number if you only have two cutters, but things get messy if you add a third or more.

Now, imagine a twist in the recipe. Instead of just asking what you can make, you ask: "If I only count every p-th cookie I make, what is the biggest number I can't reach?" This creates a new, slightly different set of numbers called a "quotient semigroup." The big question in this paper is: Is there a single, neat formula (like a magic spell) that tells us the answer for any pair of cutters and any counting step? It's like asking if there is one universal equation that predicts the impossible cookie count for every possible kitchen setup. This isn't just about cookies; it's about understanding the hidden rules of numbers, which helps in cryptography, coding theory, and even understanding how complex systems organize themselves.

The Paper's Discovery: No Magic Spell Exists

In this paper, Feihu Liu tackles a stubborn open problem: Can we write down a simple, closed-form formula for the Frobenius number of these "quotient" semigroups? Specifically, the author investigates two scenarios: one where you have two arbitrary cutters (let's call them $a$ and $b$ ) and another where the cutters are consecutive numbers (like $a$ and $a+1$ ).

The short answer is a resounding no. The paper proves that no single polynomial formula (a standard type of math equation involving powers and multiplication) can describe this number for all cases. In fact, the author shows that you can't even get away with a finite list of different formulas that switch based on the numbers you pick.

To understand how they proved this, imagine trying to fit a single, rigid plastic mold over a shape that keeps changing its size and form. The author demonstrates that as you change the numbers $a$ , $b$ , and the step $p$ , the "shape" of the answer shifts in a way that no fixed algebraic mold can capture.

Here is what the paper explicitly rules out:

No Universal Formula: There is no single polynomial equation $F(a, b, p, \text{Answer}) = 0$ that works for every possible combination of numbers.
No Finite List: You cannot solve this by making a list of, say, 10 different formulas and saying, "Use formula #1 if $a$ is prime, formula #2 if $p$ is even," etc. The paper proves that no matter how long your list is, it will eventually fail for some numbers.
No Rational Shortcut: Even if you allow fractions (rational functions) instead of just whole-number polynomials, the result is the same. There is no finite collection of these formulas that covers all cases.

How sure are they?
The paper provides a mathematical proof, not just a guess or a computer simulation. The author uses a powerful tool called Dirichlet's Theorem (which guarantees that certain patterns of numbers contain infinitely many prime numbers) to construct specific examples where the answer behaves in a way that breaks any potential formula. The logic is airtight: if a formula existed, it would have to satisfy a condition that is mathematically impossible given the infinite variety of prime numbers available.

The Twist: A Local Solution vs. A Global Failure

While the paper says "no" to a universal formula, it doesn't leave us with nothing. It finds a very specific, clever way to solve the problem if you fix one of the variables.

If you decide to keep the step size $p$ fixed (say, you always count every 5th cookie), the author shows that the answer does follow a pattern. It's not a single smooth curve, but a "quasi-polynomial." Think of this like a chameleon: if you look at the numbers where $a$ leaves a remainder of 1 when divided by 5, the answer follows one specific quadratic formula. If $a$ leaves a remainder of 2, it follows a different quadratic formula. There are at most $p$ of these different "branches."

So, for a fixed $p$ , the problem is solved! You just need to check which "branch" you are on and plug the number into the right formula. However, the paper proves that as soon as you let $p$ vary (change the step size), these branches multiply and shift in a chaotic way. The number of branches needed grows with $p$ , and the formulas themselves change so drastically that no single master formula can ever tie them all together.

The Verdict

The paper concludes that the Frobenius number for these quotient semigroups is algebraically wild. It resists being tamed by the standard tools of algebraic formulas. While we can calculate the answer for any specific case using a step-by-step algorithm (like checking remainders), the dream of a simple, all-encompassing equation is impossible. The author proves that the complexity of these numbers is intrinsic; they are too flexible to be pinned down by a finite set of polynomial rules. This result is significant because it sets a hard boundary on what is possible in number theory, showing that some patterns are simply too rich and varied to be captured by a single, neat mathematical sentence.

Technical Summary: On the Frobenius Number of Quotients of Numerical Semigroups

Problem Statement
The paper addresses the determination of explicit formulas for the Frobenius number of quotients of numerical semigroups. Given a numerical semigroup $S$ and a positive integer $p$ , the quotient $S/p = \{n \in \mathbb{N} \mid pn \in S\}$ is also a numerical semigroup. The central open problem, previously noted in literature (e.g., [24, 8, 4]), is to find a closed-form formula for the Frobenius number $g(\langle a, b \rangle / p)$ where $\gcd(a, b) = 1$ . A specific sub-case of particular interest is when the generators form consecutive integers, i.e., $S = \langle a, a+1 \rangle$ . While formulas exist for specific cases (e.g., $p=2$ , $p=5$ , or when $p$ divides $a$ ), a general closed-form expression remains elusive.

Methodology
The author employs a combination of semigroup theory, algebraic geometry, and analytic number theory to investigate the existence of polynomial formulas.

Semigroup Structure and Symmetry: The paper utilizes the property that $\langle a, b \rangle$ is a symmetric numerical semigroup. By applying a result from [28], the Frobenius number of the quotient $g(S/p)$ is expressed in terms of the Frobenius number of $S$ and the least element $\omega_p(S)$ in $S$ congruent to $g(S) \pmod p$ . For the case $S = \langle a, a+1 \rangle$ , this reduces the problem to minimizing a linear form $ia + j$ over a specific congruence class.
Arithmetic Approximation and Dirichlet's Theorem: To prove non-existence theorems, the author constructs sequences of parameters $(a, b, p)$ or $(a, p)$ where the Frobenius number follows specific linear or quadratic patterns. Lemma 4.1 utilizes Dirichlet's theorem on primes in arithmetic progressions to generate infinite sequences of primes $a$ and integers $b$ with prescribed modular properties and limiting slopes.
Algebraic Geometry (Zariski Density): The core of the non-existence proofs relies on the concept of Zariski density. The paper argues that if a polynomial relation $F(a, b, p, g) = 0$ existed, the graph of the Frobenius number would not be Zariski dense. By showing that the graph is dense (or that the required degree of any such polynomial grows unboundedly with $p$ ), the existence of such a relation is refuted.
Quasi-polynomial Analysis: For the case $S = \langle a, a+1 \rangle$ , the author derives an exact residue-class formula. This involves defining specific functions $I_p(r)$ and $J_p(r)$ based on the residue of $a$ modulo $p$ , demonstrating that for a fixed $p$ , the Frobenius number is a quadratic quasi-polynomial.

Key Contributions and Results

The paper establishes three main results regarding the algebraic nature of the Frobenius number for these quotients:

Non-existence of Uniform Polynomial Formulas (General Case):
Theorem 1.2 proves that there is no nonzero polynomial $F \in \mathbb{C}[X_1, X_2, X_3, Y]$ such that $F(a, b, p, g(\langle a, b \rangle / p)) = 0$ for all valid triples $(a, b, p)$ . Consequently, the Frobenius number cannot be represented by any finite collection of polynomial or rational formulas that are uniform in $a, b,$ and $p$ . This result is derived by showing that for a fixed prime $p$ , the Frobenius number takes on $(p-1)/2$ distinct linear forms depending on the ratio $b/a$ , and the number of such forms grows with $p$ , exceeding any fixed polynomial degree.
Quasi-polynomial Structure for Consecutive Generators:
For the specific case $S = \langle a, a+1 \rangle$ , Theorem 1.4 provides an exact formula for $G(a, p) = g(\langle a, a+1 \rangle / p)$ . The formula is:
$G(a, p) = \frac{a^2 - (I_p(r) + 1)a - (J_p(r) + 1)}{p}$
where $r = a \pmod p$ , and $I_p(r), J_p(r)$ are determined by a finite minimization procedure over residues. Theorem 1.4 concludes that for any fixed $p$ , the function $a \mapsto G(a, p)$ is a quadratic quasi-polynomial with a period dividing $p$ .
Non-existence of Polynomial Formulas (Consecutive Case):
Despite the quasi-polynomial structure for fixed $p$ , Theorem 1.5 proves that no single nonzero polynomial $F \in \mathbb{C}[X_1, X_2, X_3]$ satisfies $F(a, p, G(a, p)) = 0$ for all pairs $(a, p)$ where $a$ and $p$ are primes (or even just $p$ is prime and $a > p$ ). The proof demonstrates that the minimum degree of a polynomial relation for a fixed $p$ grows linearly with $p$ (specifically at least $\lceil p/2 \rceil$ ). Since a global polynomial must have a fixed degree, it cannot accommodate the unbounded degree requirements as $p$ varies.

Significance and Claims
The paper positions its findings as a negative answer to the open problem of finding closed-form formulas for these Frobenius numbers, inspired by Curtis's earlier work on the non-existence of formulas for $g(\langle a, b, c \rangle)$ .

Algebraic Complexity: The primary significance is the demonstration that the Frobenius number of these quotients possesses an algebraic complexity that precludes description by finite families of polynomial or rational expressions independent of the parameters.
Distinction between Local and Global: The paper highlights a contrast between local and global behavior. Locally (for fixed $p$ ), the function is well-behaved (quadratic quasi-polynomial). Globally (as $p$ varies), the function is too complex to be captured by a single algebraic relation of bounded degree.
Role of Primes: The results hold even under the restrictive condition that $a$ and $p$ are prime, utilizing Dirichlet's theorem to ensure the density of such cases.
Clarification of "Non-existence": The author explicitly notes that these results do not imply the Frobenius number is uncomputable. Explicit algorithms and formulas involving floor functions, modular arithmetic, and finite minimization (as seen in Theorem 5.3) do exist. The non-existence theorems specifically rule out polynomial or rational formulas of bounded degree that are uniform across the parameter space.

The paper concludes that while finding closed-form formulas for specific subclasses remains a valid research direction, the general case for quotients of numerical semigroups resists simple algebraic characterization.

The Great Number Hunt: Why Some Patterns Refuse to Be Tamed

The Paper's Discovery: No Magic Spell Exists

The Twist: A Local Solution vs. A Global Failure

The Verdict

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