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We present an algorithm for computing the greatest integer that is not a solution of the modular Diophantine inequality $a x mod b \leq x$ , with complexity similar to the complexity of the Euclid algorithm for computing the greatest common divisor of two integers.

On the Frobenius number of a proportionally modular Diophantine inequality.

Delgado, M., Rosales, J.C. (2006)

Portugaliae Mathematica. Nova Série

On the Frobenius number of Fibonacci numerical semigroups.

Marín, J.M., Ramírez Alfonsín, J.L., Revuelta, M.P. (2007)

Integers

On the Frobenius number of some Lucas numerical semigroups.

Ýlhan, Sedat, Kýper, Ruveyde (2008)

Acta Universitatis Apulensis. Mathematics - Informatics

On the Frobenius problem for ${a^{k}, a^{k} + 1, a^{k} + a, \dots, a^{k} + a^{k - 1}}$ .

Tripathi, Amitabha (2010)

Integers

On the limit distribution of Frobenius numbers

Andreas Strömbergsson (2012)

Acta Arithmetica

The covariety of perfect numerical semigroups with fixed Frobenius number

María Ángeles Moreno-Frías, José Carlos Rosales (2024)

Czechoslovak Mathematical Journal

Let $S$ be a numerical semigroup. We say that $h \in ℕ ∖ S$ is an isolated gap of $S$ if ${h - 1, h + 1} \subseteq S .$ A numerical semigroup without isolated gaps is called a perfect numerical semigroup. Denote by $m (S)$ the multiplicity of a numerical semigroup $S$ . A covariety is a nonempty family $𝒞$ of numerical semigroups that fulfills the following conditions: there exists the minimum of $𝒞,$ the intersection of two elements of $𝒞$ is again an element of $𝒞$ , and $S ∖ {m (S)} \in 𝒞$ for all $S \in 𝒞$ such that $S \neq min (𝒞) .$ We prove that the set $𝒫 (F) = {S : S$ is a perfect numerical semigroup with...

The Frobenius number of geometric sequences.

Ong, Darren C., Ponomarenko, Vadim (2008)

Integers

Unbounded discrepancy in Frobenius numbers.

Shallit, Jeffrey, Stankewicz, James (2011)

Integers

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