Let $p_1=2, p_2=3, p_3=5, \ldots$ be the consecutive prime numbers, $S_n$ the numerical semigroup generated by the primes not less than $p_n$ and $u_n$ the largest irredundant generator of $S_n$. We will show, that
$\bullet$ $u_n\sim3p_n$.
Similarly, for the largest integer $f_n$ not contained in $S_n$, by computational evidence we suspect that
$\bullet$ $f_n$ is an odd number for $n\geq5$ and
$\bullet$ $f_n\sim3p_n$; further
$\bullet$ $4p_n>f_{n+1}$ for $n\geq1$.
If $f_n$ is odd for large $n$, then $f_n\sim3p_n$. In case $f_n\sim3p_n$ every large even integer $x$ is the sum of two primes. If $4p_n>f_{n+1}$ for $n\geq1$, then the Goldbach conjecture holds true.
Further, Wilf's question in [12] has a positive answer for the semigroups $S_n$.
Let $0<λ\leq1$, $λ\notin\left\{\frac24, \frac27, \frac2{10}, \frac2{13}, \ldots\right\}$, be a real and $p$ a prime number, with $[p,p+λp]$ containing at least two primes. Denote by $f_λ(p)$ the largest integer which cannot be written as a sum of primes from $[p,p+λp]$. Then \[f_λ(p)\sim\left\lfloor2+\frac2λ\right\rfloor\cdot p\text{, as }p\text{ goes to infinity.}\] Further a question of Wilf about the 'Money-Changing Problem' has a positive answer for all semigroups of multiplicity $p$ containing the primes from $[p,2p]$. In particular, this holds for the semigroup generated by all primes no…
Let 0 < a < b be two relatively prime integers and let <a,b> be the numerical semigroup generated by a and b with Frobenius number g(a,b)=ab-a-b. In this note, we prove that there exists a prime number p in <a,b> with p < g(a,b) when the product ab is sufficiently large. Two related conjectures are posed and discussed as well.
We give some general results concerning the computation of the generalized Feng-Rao numbers of numerical semigroups. In the case of a numerical semigroup generated by an interval, a formula for the $r^{th}$ Feng-Rao number is obtained.
In this article, we study the quotients of numerical semigroups, generated by two coprime positive numbers, named (a,b) over d. We give formulae for the usual invariants of these semigroups, expressed in terms of continued fraction expansions and Ostrowski-like numeration of some rationals, simply related to entries a, b, d. So, we obtain quadratic complexity algorithms to compute these invariants. As a consequence, we will prove that, for these numerical semigroups, the type is always lower than the embedding dimension and deduce Wilf's property. We also consider the '' reverse problem'' : gi…
We give a simple explanation of numerical experiments of V. Arnold with two sequences of symmetric numerical semigroups, S(4,6+4k,87-4k) and S(9,3+9k,85-9k) generated by three elements. We present a generalization of these sequences by numerical semigroups S(r_1^2,r_1r_2+r_1^2k,r_3-r_1^2k), k\in{\mathbb Z}, r_1,r_2,r_3\in{\mathbb Z}^+, r_1\geq 2 and \gcd(r_1,r_2)=\gcd(r_1,r_3)=1, and calculate their universal Frobenius number Phi(r_1,r_2,r_3) for the wide range of k providing semigroups be symmetric. We show that this kind of semigroups admit also nonsymmetric representatives. We describe the …
We extend recent orbit counts for finitely generated semigroups acting on $\mathbb{P}^N$ to certain infinitely generated, polarized semigroups acting on projective varieties. We then apply these results to semigroup orbits generated by some infinite sets of unicritical polynomials.
The greatest integer that does not belong to a numerical semigroup $S$ is called the Frobenius number of $S$ and finding the Frobenius number is called the Frobenius problem. In this paper, we introduce the Frobenius problem for numerical semigroups generated by Thabit number base b and Thabit number of the second kind base b which are motivated by the Frobenius problem for Thabit numerical semigroups. Also, we introduce the Frobenius problem for numerical semigroups generated by Cunningham number and Fermat number base $b$
We derive an explicit expression for an inverse power series over the gaps values of numerical semigroups generated by two integers. It implies a set of new identities for the Hurwitz zeta function.
Consider a sequence of positive integers of the form $ca^n-d$, $n\geq 1$, where $a, c$ and $d$ are positive integers, $a>1$. For each $n\geq 1$, let $S_n$ be the submonoid of $\mathbb N$ generated by $\mathbf s_j=ca^{n+j}-d$, with $j\in\mathbb N$. We obtain a numerical semigroup $(1/e)S_n$ by dividing every element of $S_n$ by $e=\gcd(S_n)$. We characterize the embedding dimension of $S_n$ and describe a method to find the minimal generating set of $S_n$. We also show how to find the maximum element of the Apéry set ${\rm Ap}(S_n, \mathbf s_0)$, characterize the elements of ${\rm Ap}(S_n, \mat…
Let $\mathcal{A}$ be a finite set of $d\times d$ matrices with integer entries and let $m_n(\mathcal{A})$ be the maximum norm of a product of $n$ elements of $\mathcal{A}$. In this paper, we classify gaps in the growth of $m_n(\mathcal{A})$; specifically, we prove that $\lim_{n\to\infty} \log m_n(\mathcal{A})/\log n\in\mathbb{Z}_{\geqslant 0}\cup\{\infty\}.$ This has applications to the growth of regular sequences as defined by Allouche and Shallit.
We consider symmetric (not complete intersection) numerical semigroups S_6, generated by a set of six positive integers {d_1,...,d_6}, gcd(d_1,...,d_6)=1, and derive inequalities for degrees of syzygies of such semigroups and find the lower bound for their Frobenius numbers. We show that this bound may be strengthened if S_6 satisfies the Watanabe lemma.
The symmetric numerical semigroups S(F_a,F_b,F_c) and S(L_k,L_m,L_n) generated by three Fibonacci (F_a,F_b,F_c) and Lucas (L_k,L_m,L_n) numbers are considered. Based on divisibility properties of the Fibonacci and Lucas numbers we establish necessary and sufficient conditions for both semigroups to be symmetric and calculate their Hilbert generating series, Frobenius numbers and genera.
