2026/07/25 by Feihu Liu
#math.NT #math.CO #math.GR
Given a numerical semigroup S and a positive integer p, the quotient (S)/(p)=\n∈ ℕ | pn∈ S\ also forms a numerical semigroup. When S=⟨ a,b⟩ with gcd(a,b)=1, a well-known open problem is to find a closed-form formula for the Frobenius number g ((⟨ a,b⟩)/(p)), which remains open even in the special case b=a+1. Inspired by Curtis's theorem on the non-existence of polynomial formulas for the Frobenius number g(⟨ s1,s2,s3⟩), we provide a negative answer to this open problem in a certain sense. Concretely, we obtain the following three main results. (i): The Frobenius number g ((⟨ a,b⟩)/(p)) cannot be represented, uniformly in a,b,p, by any finite collection of polynomial (or rational) formulas. (ii): For each fixed p, the function a↦ g ((⟨ a,a+1⟩)/(p)) is a quadratic quasi-polynomial with period dividing p. (iii): There is no nonzero polynomial F∈ ℂ[X1,X2,X3] satisfying F(a,p,g ((⟨ a,a+1⟩)/(p)))=0 for all primes a,p with 2<p<a; the same conclusion already holds if only p is required to be prime and a ranges over all integers greater than p. While (iii) is stronger than (i), the proofs of the two results reveal different insights. Dirichlet's theorem on primes in arithmetic progressions plays a crucial role in our arguments.