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A numerical semigroup S is dense if for all $$s\in S\backslash \{0\}$$ we have $$\left\{ s-1,s+1\right\} \cap S\ne \emptyset $$ . We give algorithms to compute the whole set of dense numerical semigroups with fixed genus, Frobenius number and multiplicity. Furthermore, we solve the Frobenius problem for dense numerical semigroups with embedding dimension three.

In this note we compare two measures of the complexity of a class $\mathcal F$ of Boolean functions studied in (unconditional) pseudorandomness: $\mathcal F$'s ability to distinguish between biased and uniform coins (the coin problem), and the norms of the different levels of the Fourier expansion of functions in $\mathcal F$ (the Fourier growth). We show that for coins with low bias $\varepsilon = o(1/n)$, a function's distinguishing advantage in the coin problem is essentially equivalent to $\varepsilon$ times the sum of its level $1$ Fourier coefficients, which in particular shows that known level $1$ and total influence bounds for some classes of interest (such as constant-width read-once branching programs) in fact follow as a black-box from the corresponding coin theorems, thereby simplifying the proofs of some known results in the literature. For higher levels, it is well-known that Fourier growth bounds on all levels of the Fourier spectrum imply coin theorems, even for large $\varepsilon$, and we discuss here the possibility of a converse.

Posebni problem razmene denarja je le ena od variacij klasičnega problema razmene denarja, ki pa je prav tako le ena od variacij najširšega optimizacijskega problema v tej zgodbi – preprostega problema nahrbtnika. Najožja izmed teh variacij, posebni problem razmene denarja, sprašuje po vrednostih kovancev $a_1, a_2, ldots , a_t$, za katere obstaja natanko ena razmena z najmanjšim možnim skupnim številom kovancev teh vrednosti, ki jih uporabimo za razmeno, za vsako vsoto denarja, ki se jo s kovanci teh vrednosti da razmenjati. Predstavili bomo rešitev problema v primeru dveh kovancev različnih vrednosti ter nekaj metod za iskanje (oz. preverjanje ustreznosti) rešitev v primeru treh kovancev različnih vrednosti, ki ustrezajo določenim pogojem. The special money-changing problem is only one of the many variations of the classical money-changing problem, which is also only one of the many variations of the widest optimisation problem in this story – the knapsack problem. The narrowest of these variations, the special money-changing problem, asks for what denominations of money $a_1, a_2, ldots , a_t$, is there exactly one way to make change, using the fewest number of coins possible, for every amount for which change can be made using only coins of these denominations. We will provide a solution to this problem in the case with coins of two different denominations and a few methods for finding (or testing the correctness of) solutions in the case with coins of three different denominations, which all meet certain conditions.

The (Diophantine) Frobenius problem is a well-known NP-hard problem (also called the stamp problem or the chicken nugget problem) whose origins lie in the realm of combinatorial number theory. In this paper we present an algorithm which solves it, via a translation into a QUBO problem of the so-called Apery set of a numerical semigroup. This algorithm was specifically designed to run in an adiabatic quantum computer (a D-Wave 2X machine), and the performance problems for this precise setting are also discussed.

Consider the problem of computing the majority of a stream of $n$ i.i.d. uniformly random bits. This problem, known as the coin problem, is central to a number of counting problems in different data stream models. We show that any streaming algorithm for solving this problem with large constant advantage must use $\Omega(\log n)$ bits of space. We extend our lower bound to proving tight lower bounds for solving multiple, randomly interleaved copies of the coin problem, as well as for solving the OR of multiple copies of a variant of the coin problem. Our proofs involve new measures of information complexity that are well-suited for data streams. We use these lower bounds to obtain a number of new results for data streams. In each case there is an underlying $d$ dimensional vector $x$ with additive updates to its coordinates given in a stream of length $m$ . The input streams arising from our coin lower bound have nice distributional properties, and consequently for many problems for which we only had lower bounds in general turnstile streams, we now obtain the same lower bounds in more natural models, such as the bounded deletion model, in which $\Vert x\Vert_{2}$ never drops by a constant fraction of what it was earlier, or in the random order model, in which the updates are ordered randomly. In particular, in the bounded deletion model, we obtain nearly tight lower bounds for approximating $\Vert x\Vert_{\infty}$ up to additive error $\frac{1}{\sqrt{k}}\Vert x\Vert_{2}$ , approximating $\Vert x\Vert_{2}$ up to a multiplicative ( $1+\epsilon$ ) factor (resolving a question of Jayaram and Woodruff in PODS 2018), and solving the Point Query and $\ell_{2}$ -Heavy Hitters Problems. In the random order model, we also obtain new lower bounds for the Point Query and $\ell_{2}$ -Heavy Hitters Problems. We also give new algorithms complementing our lower bounds and illustrating the tightness of the models we consider, including an algorithm for approximating $\Vert x\Vert_{\infty}$ up to additive error $\frac{1}{\sqrt{k}}\Vert x\Vert_{2}$ in turnstile streams (resolving a question of Cormode in a 2006 IITK Workshop), and an algorithm for finding $\ell_{2}$ -heavy hitters in randomly ordered insertion streams (which for random order streams, resolves a question of Nelson in a 2018 Warwick Workshop).

We provide an expression for the conductor $c(\Delta)$ of a semimodule $\Delta$ of a numerical semigroup $\Gamma$ with two generators in terms of the syzygy module of $\Delta$ and the generators of the semigroup. In particular, we deduce that the difference between the conductor of the semimodule and the conductor of the semigroup is an element of $\Gamma$, as well as a formula for $c(\Delta)$ in terms of the dual semimodule of $\Delta$., Comment: Proof of Theorem 3.1 has been clarified. Some typos corrected along the whole paper. To Appear in Semigroup Forum. arXiv admin note: text overlap with arXiv:2011.01690. text overlap with arXiv:2011.01690

Hegedűs’s lemma is the following combinatorial statement regarding polynomials over finite fields. Over a field F of characteristic p > 0 and for q a power of p, the lemma says that any multilinear polynomial P∈ F[x 1,…,x n ] of degree less than q that vanishes at all points in {0,1} n of Hamming weight k∈ [q,n−q] must also vanish at all points in {0,1} n of weight k + q. This lemma was used by Hegedűs (2009) to give a solution to Galvin’s problem, an extremal problem about set systems; by Alon, Kumar and Volk (2018) to improve the best-known multilinear circuit lower bounds; and by Hrubes, Ramamoorthy, Rao and Yehudayoff (2019) to prove optimal lower bounds against depth-2 threshold circuits for computing some symmetric functions. In this paper, we formulate a robust version of Hegedűs’s lemma. Informally, this version says that if a polynomial of degree o(q) vanishes at most points of weight k, then it vanishes at many points of weight k+q. We prove this lemma and give the following three different applications. 1. Degree lower bounds for the coin problem: The δ-Coin Problem is the problem of distinguishing between a coin that is heads with probability ((1/2) + δ) and a coin that is heads with probability 1/2. We show that over a field of positive (fixed) characteristic, any polynomial that solves the δ-coin problem with error e must have degree Ω(1/δlog(1/e)), which is tight up to constant factors. 2. Probabilistic degree lower bounds: The Probabilistic degree of a Boolean function is the minimum d such that there is a random polynomial of degree d that agrees with the function at each point with high probability. We give tight lower bounds on the probabilistic degree of every symmetric Boolean function over positive (fixed) characteristic. As far as we know, this was not known even for some very simple functions such as unweighted Exact Threshold functions, and constant error. 3. A robust version of the combinatorial result of Hegedűs (2009) mentioned above.

U ovom radu opisan je postupak rješavanja McNuggetovog problema za proizvoljne veličine kutija. Oblikovani su algoritmi za rješavanje 3 potproblema: pronalaženje Frobeniusovog broja, odluka je li broj Mcnuggetov i prikaz McNuggetovog broja preko veličina kutija koje su dane. Svi algoritmi implementirani su u programskom jeziku Python. Rezultati njihovog izvršavanja su analizirani i uspoređeni., This paper describes a procedure for solving a McNugget problem for arbitrary box sizes. Algorithms have been devised to solve 3 subproblems: finding a Frobenius number, deciding whether the number is McNugget number, and displaying a McNugget number with the given box sizes. All algorithms are implemented in Python programming language. The results of their execution are analyzed and compared.

Computational pseudorandomness studies the extent to which a random variable $\bf{Z}$ looks like the uniform distribution according to a class of tests $\cal{F}$. Computational entropy generalizes computational pseudorandomness by studying the extent which a random variable looks like a \emph{high entropy} distribution. There are different formal definitions of computational entropy with different advantages for different applications. Because of this, it is of interest to understand when these definitions are equivalent. We consider three notions of computational entropy which are known to be equivalent when the test class $\cal{F}$ is closed under taking majorities. This equivalence constitutes (essentially) the so-called \emph{dense model theorem} of Green and Tao (and later made explicit by Tao-Zeigler, Reingold et al., and Gowers). The dense model theorem plays a key role in Green and Tao's proof that the primes contain arbitrarily long arithmetic progressions and has since been connected to a surprisingly wide range of topics in mathematics and computer science, including cryptography, computational complexity, combinatorics and machine learning. We show that, in different situations where $\cal{F}$ is \emph{not} closed under majority, this equivalence fails. This in turn provides examples where the dense model theorem is \emph{false}., Comment: 19 pages; to appear in ITCS 2021

Let X m,ε be the distribution over m bits X 1 ,…, X m where the X i are independent and each X i equals 1 with probability (1− ε )/2 and 0 with probability (1 − ε )/2. We consider the smallest value ε * of ε such that the distributions X m, ε and X m, 0 can be distinguished with constant advantage by a function f : {0,1} m → S , which is the product of k functions f 1 , f 2 ,…, f k on disjoint inputs of n bits, where each f i : {0,1} n → S and m = nk . We prove that ε * = Θ(1/√ n log k ) if S = [−1,1], while ε * = Θ(1/√ nk ) if S is the set of unit-norm complex numbers.

Producción Científica, This work generalises the short resolution given by Pisón Casares [‘The short resolution of a lattice ideal’, Proc. Amer. Math. Soc. 131(4) (2003), 1081–1091] to any affine semigroup. We give a characterisation of Apéry sets which provides a simple way to compute Apéry sets of affine semigroups and Frobenius numbers of numerical semigroups. We also exhibit a new characterisation of the Cohen–Macaulay property for simplicial affine semigroups., Ministerio de Economía, Industria y Competitividad - Fondo Europeo de Desarrollo Regional (projects MTM2012-36917-C03-01 / MTM2015-65764-C3-1-P)

We deduce the law of nonstationary recursion which makes it possible, for given a primitive set A = {a 1,...,a k }, k > 2, to construct an algorithm for finding the set of the numbers outside the additive semigroup generated by A. In particular, we obtain a new algorithm for determining the Frobenius numbers g(a 1,...,a k ). The computational complexity of these algorithms is estimated in terms of bit operations. We propose a two-stage reduction of the original primitive set to an equivalent primitive set that enables us to improve complexity estimates in the cases when the two-stage reduction leads to a substantial reduction of the order of the initial set.

Let [Formula: see text] be two positive integers such that [Formula: see text] and [Formula: see text] the numerical semigroup generated by [Formula: see text]. Then [Formula: see text] is the Thabit numerical semigroup introduced by J. C. Rosales, M. B. Branco and D. Torrão. In this paper, we give formulas for computing the Frobenius number, the genus and the embedding dimension of [Formula: see text].

We study the solvability of linear diophantine equations in two variables within the context of Beatty sequences.

Frobenius problem and its many generalizations have been extensively studied in several areas of mathematics. We study semigroups of totally positive algebraic integers in totally real number fields, defining analogues of the Frobenius numbers in this context. We use a geometric framework recently introduced by Aliev, De Loera and Louveaux to produce upper bounds on these Frobenius numbers in terms of a certain height function. We discuss some properties of this function, relating it to absolute Weil height and obtaining a lower bound in the spirit of Lehmer's conjecture for algebraic vectors satisfying some special conditions. We also use a result of Borosh and Treybig to obtain bounds on the size of representations and number of elements of bounded height in such positive semigroups of totally real algebraic integers., 13 pages, to appear in Moscow Journal of Combinatorics and Number Theory

In this work we prove the first Fixed-depth Size-Hierarchy Theorem for uniform AC0[⊕]. In particular, we show that for any fixed d, the class Cd,k of functions that have uniform AC0[⊕] formulas of depth d and size nk form an infinite hierarchy. We show this by exhibiting the first class of explicit functions where we have nearly (up to a polynomial factor) matching upper and lower bounds for the class of AC0[⊕] formulas. The explicit functions are derived from the δ-Coin Problem, which is the computational problem of distinguishing between coins that are heads with probability (1+δ)/2 or (1−δ)/2, where δ is a parameter that is going to 0. We study the complexity of this problem and make progress on both upper bound and lower bound fronts. Upper bounds. For any constant d≥ 2, we show that there are explicit monotone AC0 formulas (i.e. made up of AND and OR gates only) solving the δ-coin problem that have depth d, size exp(O(d(1/δ)1/(d−1))), and sample complexity (i.e. number of inputs) poly(1/δ). This matches previous upper bounds of O’Donnell and Wimmer (ICALP 2007) and Amano (ICALP 2009) in terms of size (which is optimal) and improves the sample complexity from exp(O(d(1/δ)1/(d−1))) to poly(1/δ). Lower bounds. We show that the above upper bounds are nearly tight (in terms of size) even for the significantly stronger model of AC0[⊕] formulas (which are also allowed NOT and Parity gates): formally, we show that any AC0[⊕] formula solving the δ-coin problem must have size exp(Ω(d(1/δ)1/(d−1))). This strengthens a result of Shaltiel and Viola (SICOMP 2010), who prove a exp(Ω((1/δ)1/(d+2))) lower bound for AC0[⊕], and a lower bound of exp(Ω((1/δ)1/(d−1))) shown by Cohen, Ganor and Raz (APPROX-RANDOM 2014) for the class 0. The upper bound is a derandomization involving a use of Janson’s inequality and classical combinatorial designs. The lower bound involves proving an optimal degree lower bound for polynomials over 2 solving the δ-coin problem.

Editors Preface SECTION ONE: Solitaire Games with Toys Solving Instant Insanity Robert E. Levin The Mayblox Problem Margaret A. Farrell A Solitaire Game and Its Relation to a Finite Field N. G. de Bruijn Triangular Puzzle Peg Irvin Roy Hentzel Parity and Centerness Applied to the SOMA Cube Michael J. Whinihanand Charles W. Trigg The Tower of Brahma Revisited Ted Roth Tower of Hanoi with More Pegs Brother Alfred Brousseau SECTION TWO: Competitive Games Compound Games with Counters Cedric A. B. Smith The Game of SIM Gustavus J. Simmons Some Investigations into the Game of SIM A. P. DeLoach SIM as a Game of Chance W. W. Funkenbusch SIM on a Desktop Calculator John N. Nairn and A. B. Sperry A Winning Strategy for SIM E. M. Rounds and S. S. Yau The Graph of Positions for the Game of SIM G. L. O'Brien Dots and Squares Ernest R. Ranucci An Analysis of "Square It" Thomas S. Briggs Dots and Triangles Joseph Viggiano Dots and Cubes Everett V. Jackson A Winning Opening in Reverse Hex Ronald Evans SECTION THREE: Solitaire Games Arrows and Circuits Brian R. Barwell Knight Interchanges: 1 Robert E. Parkin Knight Interchanges: 2 Ted Roth The Stacked Playing Cards Robert E. Parkin Extension of the Chain-Cutting Problem Donald R. Byrkit and William M. Walters, Jr. The "12 + 1" False Coin Problem M. H. Greenblatt BONUS SECTION: The Four-Color Problem The Mathematics of Map Coloring H. S. M. Coxeter Every Planar Map is Four Colorable Kenneth Appel and Wolfgang Haken

In this paper, we give formulas for the embedding dimension, the Frobenius number, the type and the genus for a numerical semigroups generated by the Mersenne numbers greater than or equal to a given Mersenne number.

Asymptotic formulae for the mean values of various characteristics of the additive semigroup generated by three positive integers are obtained theoretically, the first of which is a formula for the number of integers not belonging to this semigroup. A numerical experiment is described which validates the results obtained. Bibliography: 21 titles.

For positive and relatively prime set of integers A, let Γ ( A ) denote the set of integers that is formed by taking nonnegative integer linear combinations of integers in A. Then there are finitely many positive integers that do not belong to Γ ( A ) . For A = { a , h a + d , h a + b d , h a + b 2 d , … , h a + b k d } , gcd ⁡ ( a , d ) = 1 , we determine the largest integer g ( A ) that does not belong to Γ ( A ) , and the number of integers n ( A ) that does not belong to Γ ( A ) , both for all sufficiently large values of d. This extends a result of Selmer, and corrects a result of Hofmeister, both given in special cases.

For the Sierpinski-like sponge in R n , we obtain the Hausdorff dimension of the set of differentiable points such that there exists an ( n − 1 ) -rectifiable subsurface which has a tangent hyperplane at the point, using the techniques such as multi dynamical systems, geometric types of slices and restricted Frobenius problem. Furthermore, we can use the dimension of the set of differentiable points, as a bilipschitz invariant, to classify Sierpinski-like sponges in terms of bilipschitz equivalence.

Our goal is to convince the readers that the theory of complex normal surface singularities can be a powerful tool in the study of numerical semigroups, and, in the same time, a very rich source of interesting affine and numerical semigroups. More precisely, we prove that the strongly flat semigroups, which satisfy the maximality property with respect to the Diophantine Frobenius problem, are exactly the numerical semigroups associated with negative definite Seifert homology spheres via the possible 'weights' of the generic $S^1$-orbit. Furthermore, we consider their generalization to the Seifert rational homology sphere case and prove an explicit (up to a Laufer computation sequence) formula for their Frobenius number. The singularities behind are the weighted homogeneous ones, whose several topological and analytical properties are exploited., 23 pages

Let S be a map from a language L to the integers satisfying S ( v w ) = S ( v ) + S ( w ) for all v , w ∈ L . The classical Frobenius problem asks whether the complement of S ( L ) in the natural numbers will be infinite or finite, and in the latter case the value of the largest element in this complement. This is also known as the ‘coin-problem’, and L is the full language consisting of all words over a finite alphabet. We solve the Frobenius problem for the golden mean language, any Sturmian language and the Thue–Morse language. We also consider two-dimensional embeddings.

Let n be a positive integer and T ( n ) the numerical semigroup generated by { 3.2 n + i − 1 | i ∈ N } . In this paper, we give formulas for the Frobenius number, the gender, the embedding dimension and the type of T ( n ) .

A repunit is a number consisting of copies of the single digit 1. The set of repunits in base b is \(\big \{\frac{b^n-1}{b-1} ~|~ n\in {\mathbb N}\backslash \{0\}\big \}\). A numerical semigroup S is a repunit numerical semigroup if there exist integers \(b\in {\mathbb N}\backslash \left\{ 0,1\right\} \) and \(n\in {\mathbb N}\backslash \left\{ 0\right\} \) such that \(S=\big \langle \big \{\frac{b^{n+i}-1}{b-1} ~|~ i\in {\mathbb N}\big \}\big \rangle \). In this work, we give formulas for the embedding dimension, the Frobenius number, the type and the genus for a repunit numerical semigroup.

We show that Ambro-Kawamata's non-vanishing conjecture holds true for a quasi-smooth WCI X which is Fano or Calabi-Yau, i.e. we prove that, if H is an ample Cartier divisor on X, then |H| is not empty. If X is smooth, we further show that the general element of |H| is smooth. We then verify Ambro-Kawamata's conjecture for any quasi-smooth weighted hypersurface. We also verify Fujita's freeness conjecture for a Gorenstein quasi-smooth weighted hypersurface. For the proofs, we introduce the arithmetic notion of regular pairs and enlighten some interesting connection with the Frobenius coin problem., 27 pages. Revised version to appear in Algebra and Number Theory

We solve the Frobenius coin problem in the Gaussian integers when the coin set is comprised of two conjugate primes. We also provide preliminary results toward a general solution for the two-coin problem in the Gaussian integers.

The 8-coin insight problem requires the problem solver to move 2 coins so that each coin touches exactly 3 others. Ormerod, MacGregor, and Chronicle (2002) explained differences in task performance across different versions of the 8-coin problem using the availability of particular moves in a 2-dimensional search space. We explored 2 further explanations by developing 6 new versions of the 8-coin problem in order to investigate the influence of grouping and self-imposed constraints on solutions. The results identified 2 sources of problem difficulty: first, the necessity to overcome the constraint that a solution can be found in 2-dimensional space and, second, the necessity to decompose perceptual groupings. A detailed move analysis suggested that the selection of moves was driven by the established representation rather than the application of the appropriate heuristics. Both results support the assumptions of representational change theory (Ohlsson, 1992).

We characterize the finite sets S of words such that that the iterated shuffle of S is co-finite and we give some bounds on the length of a longest word not in the iterated shuffle of S., 16 pages

This paper provides a formula for the minimal relations and the Frobenius number of a numerical semigroup minimally generated by three pairwise coprime positive integers., Minor corrections

Given an integral d×n matrix A, the well-studied affine semigroup Sg(A)={b:Ax=b, x∈Zn,x≥0} can be stratified by the number of lattice points inside the parametric polyhedra PA(b)={x:Ax=b,x≥0}. Such families of parametric polyhedra appear in many areas of combinatorics, convex geometry, algebra and number theory. The key themes of this paper are: (1) A structure theory that characterizes precisely the subset Sg≥k(A) of all vectors b∈ Sg(A) such that PA(b)∩Zn has at least k solutions. We demonstrate that this set is finitely generated, it is a union of translated copies of a semigroup which can be computed explicitly via Hilbert bases computations. Related results can be derived for those right-hand-side vectors b for which PA(b)∩Zn has exactly k solutions or fewer than k solutions. (2) A computational complexity theory. We show that, when n, k are fixed natural numbers, one can compute in polynomial time an encoding of Sg≥k(A) as a multivariate generating function, using a short sum of rational functions. As a consequence, one can identify all right-hand-side vectors of bounded norm that have at least k solutions. (3) Applications and computation for the k-Frobenius numbers. Using Generating functions we prove that for fixed n,k the k-Frobenius number can be computed in polynomial time. This generalizes a well-known result for k=1 by R. Kannan. Using some adaptation of dynamic programming we show some practical computations of k-Frobenius numbers and their relatives.

If S is a numerical semigroup with embedding dimension equal to three whose minimal generators are pairwise relatively prime numbers, then S =ha;b;cb dai with a;b;c;d positive integers such that gcd(a;b) = gcd(a;c) = gcd(b;d) = 1, c2f2;:::;a 1g, anda < b < cb da. In this paper we give formulas, in terms ofa;b;c;d, for the genus, the Frobenius number, and the set of pseudo-Frobenius numbers ofha;b;cb dai in the case in which the interval a ; b d contains some integer.

A (non-circular) de Bruijn sequence w of order n is a word such that every word of length n appears exactly once in w as a factor. In this paper, we generalize the concept to different settings: the multi-shift de Bruijn sequence and the pseudo de Bruijn sequence. An m-shift de Bruijn sequence of order n is a word such that every word of length n appears exactly once in w as a factor that starts at a position im + 1 for some integer i ≥ 0. A pseudo de Bruijn sequence of order n with respect to an antimorphic involution θ is a word such that for every word u of length n the total number of appearances of u and θ(u) as a factor is one. We show that the number of m-shift de Bruijn sequences of order n is an!a(m-n)(an-1) for 1 ≤ n ≤ m and is (am!)an-m for 1 ≤ m ≤ n, where a is the size of the alphabet. We provide two algorithms for generating a multi-shift de Bruijn sequence. The multi-shift de Bruijn sequence is important for solving the Frobenius problem in a free monoid. We show that the existence of pseudo de Bruijn sequences depends on the given alphabet and antimorphic involution, and obtain formulas for the number of such sequences in some particular settings.

In this note we observe that the Frobenius number and therefore the conductor of a numerical semigroup can be obtained from the maximal socle degree of the quotient of the corresponding semigroup algebra by the ideal generated by the biggest generator of the semigroup., Comment: Some typos in the introduction have been corrected

We examine a version of the Universal Number Partition Problem with a divisibility property referred to as the Universal Shelf Packing Problem (USPP). We show that if a shelf length is a product of powers of two primes the USPP is always partitionable. In the case where the shelf length is a product of three distinct primes we propose an efficient scheme to determine when such a case is not partitionable. We also show that a shelf length that is a product of powers of four or more primes always has at least one partition failure. Our analysis uses elementary number theory, known results related to the linear Diophantine Frobenius problem, and a new result on Frobenius gaps.

We provide a variety of results concerning the problem of determining maximal vectors [math] such that the Diophantine system [math] has no solution: conditions for the existence of [math] , conditions for the uniqueness of [math] , bounds on [math] , determining [math] explicitly in several important special cases, constructions for [math] , and a reduction for [math] .

In this paper, we study the gender, Frobenius number and pseudo-Frobenius number for numerical semigroups with multiplicity four, embedding dimension three and minimal generators pairwise relatively prime.