Related papers: A Frobenius problem suggested by prime $k$-tuplets…
A Diophantine $m$-tuple is a set of $m$ distinct integers such that the product of any two distinct elements plus one is a perfect square. In this paper we study the extensibility of a Diophantine triple $\{k-1, k+1, 16k^3-4k\}$ in Gaussian…
An s-tuple of positive integers are k-wise relatively prime if any k of them are relatively prime. Exact formula is obtained for the probability that s positive integers are k-wise relatively prime.
The matrix representation of the set $\Delta({\bf d}^3)$, ${\bf d}^3=(d_1,d_2, d_3)$, of the integers which are unrepresentable by $d_1,d_2,d_3$ is found. The diagrammatic procedure of calculation of the generating function $\Phi({\bf…
We connect Dedekind sums and some formulas for numerical semigroups.
In this paper, we investigate the umbral representation of the Fubini polynomials $F_{x}^{n}:=F_{n}(x)$ to derive some properties involving these polynomials. For any prime number $p$ and any polynomial $f$ with integer coefficients, we…
Recently, Drema and Saikia (2023) proved several congruences modulo powers of 2 and 3 for overpartition triples with odd parts. We extend their list substantially. We prove several congruences modulo powers of 2 for overpartition k-tuples…
In this work we present a new class of numerical semigroups called GSI-semigroups. We see the relations between them and others families of semigroups and we give explicitly their set of gaps. Moreover, an algorithm to obtain all the…
We give some new examples of Frobenius objects in the category of sets and relations $\textbf{Rel}$. One example is a groupoid with a twisted counit. Another example is the set of conjugacy classes of a group. We also classify Frobenius…
Numerical semigroups have been extensively studied throughout the literature, and many of their invariants have been characterized. In this work, we generalize some of the most important results about symmetry, pseudo-symmetry, or…
The twin primes conjecture is a very old problem. Tacitly it is supposed that the primes it deals with are finite. In the present paper we consider three problems that are not related to finite primes but deal with infinite integers. The…
We study statistical properties of random numerical semigroups of a given genus. We analyze the graph of a typical numerical semigroup, understood as a function from $\mathbb{N}$ to $\mathbb{N}$. If $S$ is a numerical semigroup of genus…
In this paper, we investigate the computational complexity of the knapsack problem and subset sum problem for the following tropical algebraic structures. We consider the semigroup of square matrices of size $k \times k$ with non-negative…
We formulate some conjectures that relates semisimple Frobenius manifolds, their spectral curves and integrable hierarchies.
The pseudo-Frobenius numbers of a numerical semigroup are those gaps of the numerical semigroup that are maximal for the partial order induced by the semigroup. We present a procedure to detect if a given set of integers is the set of…
Grayson, developing ideas of Quillen, has made computations of the K-theory of "semi-linear endomorphisms". In the present text we develop a technique to compute these groups in the case of Frobenius semi-linear actions. The main idea is to…
Let a, k, h, c be positive integers and d a non zero integer. Recall that a numerical generalized almost arithmetic semigroup S is a semigroup minimally generated by relatively prime positive integers a, ha + d, ha + 2d, . . . , ha + kd, c,…
A Grobner basis-based algorithm for solving the Frobenius Instance Problem is presented, and this leads to an algorithm for solving the Frobenius Problem that can handle numbers with thousands of digits. Connections to irreducible…
In this work we introduce the notion of almost-symmetry for generalized numerical semigroups. In addition to the main properties occurring in this new class, we present several characterizations for its elements. In particular we show that…
Let $\alpha_1,\ldots,\alpha_r$ be algebraic numbers in a number field $K$ generating a subgroup of rank $r$ in $K^\times$. We investigate under GRH the number of primes $\mathfrak p$ of $K$ such that each of the orders of…
We derive the lower bound for Frobenius number of symmetric (not complete intersection) semigroups generated by four elements.