On formulas for the Frobenius number of a numerical semigroup.

Curtis, Frank

doi:10.7146/math.scand.a-12330

Cited by 85 publications

(69 citation statements)

References 2 publications

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“…We note that if p = 3, then G 2 = G 0 and κ + λ ≥ 3, so that parts (i) and (ii) of Theorem 2.1 imply that g = G 0 (and hence also g = G 2 , as in part (iii)). For p > 3, the DIOPHANTINE FROBENIUS PROBLEM RELATED TO RIEMANN SURFACES 503 integer q = (p − 3)p + 1, if prime, is the largest such that κ + λ < p. Hence, we obtain an easy corollary.…”

Section: The Main Results Define the Functionmentioning

confidence: 99%

A Diophantine Frobenius Problem Related to Riemann Surfaces

O’Sullivan

Weaver

2011

Glasgow Math. J.

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Abstract. We obtain sharp upper and lower bounds on a certain four-dimensional Frobenius number determined by a prime pair (p, q), 2 < p < q, including exact formulae for two infinite subclasses of such pairs. Our work is motivated by the study of compact Riemann surfaces which can be realised as semi-regular pq-fold coverings of surfaces of lower genus. In this context, the Frobenius number is (up to an additive translation) the largest genus in which no surface is such a covering. In many cases it is also the largest genus in which no surface admits an automorphism of order pq. The general t-dimensional Frobenius problem (t ≥ 3) is NP-hard, and it may be that our restricted problem retains this property.

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Section: The Main Results Define the Functionmentioning

confidence: 99%

A Diophantine Frobenius Problem Related to Riemann Surfaces

O’Sullivan

Weaver

2011

Glasgow Math. J.

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“…No general formula has been found so far for the case p ≥ 3. Moreover, as Curtis shows in [1], there is no closed formula of a certain type for p = 3. If we focus our attention on this case, then we can think of F as a correspondence that maps three relatively prime integers n 1 , n 2 , n 3 to a nonnegative integer F(n 1 , n 2 , n 3 ).…”

Section: Introductionmentioning

confidence: 99%

Every positive integer is the Frobenius number of a numerical semigroup with three generators

Rosales¹,

García-Sánchez²,

García-García³

2004

MATH. SCAND.

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“…In actual, unlike Sylvester's elegant formula, there are no polynomial solutions for more than two integers [1]. In general, it is NP-hard to find the Frobenius numbers for the number of given integers [4].…”

Section: Introductionmentioning

confidence: 99%