2015
DOI: 10.1007/s00209-015-1412-0
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On a conjecture by Wilf about the Frobenius number

Abstract: Given coprime positive integers a 1 < · · · < a d , the Frobenius number F is the largest integer which is not representable as a non-negative integer combination of the a i . Let g denote the number of all non-representable positive integers: Wilf conjectured that d ≥ F+1 F+1−g . We prove that for every fixed value of a 1 d the conjecture holds for all values of a 1 which are sufficiently large and are not divisible by a finite set of primes. We also propose a generalization in the context of one-dimensional … Show more

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Cited by 22 publications
(21 citation statements)
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“…In 1978, Wilf proposed a conjecture related to the Diophantine Frobenius Problem ( [14]) that claims that the inequality n(S) · e(S) ≥ Fb(S) + 1 is true for every numerical semigroup. This conjecture still remains open, and it has become an important part of the Theory of Numerical Semigroup, for instance it has been studied in [2], [6], [7], [9], [11] and the references therein. Most of these papers describe families of semigroups satisfying Wilf's conjecture.…”
Section: Introductionmentioning
confidence: 99%
“…In 1978, Wilf proposed a conjecture related to the Diophantine Frobenius Problem ( [14]) that claims that the inequality n(S) · e(S) ≥ Fb(S) + 1 is true for every numerical semigroup. This conjecture still remains open, and it has become an important part of the Theory of Numerical Semigroup, for instance it has been studied in [2], [6], [7], [9], [11] and the references therein. Most of these papers describe families of semigroups satisfying Wilf's conjecture.…”
Section: Introductionmentioning
confidence: 99%
“…Let z ∈ H, and assume z = x + y with x, y integers such that 1 ≤ x ≤ y. We need show (13) x ∈ H or y ∈ H.…”
Section: A Lower Bound On N Gmentioning
confidence: 99%
“…In fact, we shall almost always obtain the stronger estimate W (S) 1. See [17,Question 8] or [9, Page 2] for a conjecture about the rare occurrences of the equality W (S) = 0.…”
Section: Proof Of Main Theoremmentioning
confidence: 99%
“…Wilf's conjecture, as it is now known, has been verified in several cases, including when |P | 3, or c 3m, or m 18, or |L| 12, or |P | m/2. See Delgado [5] for an extensive recent survey of partial results on Wilf's conjecture, and [1,2,4,9,11,13,14,15,17,20,21,22] for some relevant papers. The verification in case |P | m/2 is due to Sammartano [20] in 2012.…”
Section: Introductionmentioning
confidence: 99%