scholarly journals The Polynomial Part of a Restricted Partition Function Related to the Frobenius Problem

10.37236/1592 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Matthias Beck ◽  
Ira M. Gessel ◽  
Takao Komatsu
Keyword(s):  
Partition Function ◽  
Explicit Formula ◽  
Nonnegative Integer ◽  
Frobenius Problem ◽  
Restricted Partition ◽  
Integer Solutions ◽  
Positive Integers

Given a set of positive integers $ A = \{ a_{1} , \dots , a_{n} \} $, we study the number $ p_{A} (t) $ of nonnegative integer solutions $ \left( m_{1} , \dots , m_{n} \right) $ to $ \sum_{j=1}^{n} m_{j} a_{j} = t $. We derive an explicit formula for the polynomial part of $p_A$.

scholarly journals On the Multiplicity of a Proportionally Modular Numerical Semigroup

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Ze Gu
Keyword(s):  
Positive Integer ◽  
Nonnegative Integer ◽  
Numerical Semigroup ◽  
Diophantine Inequality ◽  
Integer Solutions ◽  
Positive Integers

A proportionally modular numerical semigroup is the set S a , b , c of nonnegative integer solutions to a Diophantine inequality of the form a x   mod   b ≤ c x , where a , b , and c are positive integers. A formula for the multiplicity of S a , b , c , that is, m S a , b , c = k b / a for some positive integer k , is given by A. Moscariello. In this paper, we give a new proof of the formula and determine a better bound for k . Furthermore, we obtain k = 1 for various cases and a formula for the number of the triples a , b , c such that k ≠ 1 when the number a − c is fixed.

On the Ramanujan-Nagell type Diophantine equation \(Dx^2+k^n=B\)

2021 ◽  
Vol 56 (2) ◽  
pp. 263-270
Author(s):  
Zhongfeng Zhang ◽  
◽  
Alain Togbé ◽  
Keyword(s):  
Diophantine Equation ◽  
Nonnegative Integer ◽  
Integer Solutions ◽  
Positive Integers

In this paper, we prove that the Ramanujan-Nagell type Diophantine equation \(Dx^2+k^n=B\) has at most three nonnegative integer solutions \((x, n)\) for \(k\) a prime and \(B, D\) positive integers.

scholarly journals The Parametric Frobenius Problem

10.37236/5112 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Bjarke Hammersholt Roune ◽  
Kevin Woods
Keyword(s):  
Nonnegative Integer ◽  
Frobenius Number ◽  
Linear Functions ◽  
Frobenius Problem ◽  
Number Of Generators ◽  
Positive Integers ◽  
Parametric Version

Given relatively prime positive integers $a_1,\ldots,a_n$, the Frobenius number is the largest integer that cannot be written as a nonnegative integer combination of the $a_i$. We examine the parametric version of this problem: given $a_i=a_i(t)$ as functions of $t$, compute the Frobenius number as a function of $t$. A function $f:\mathbb{Z}_+\rightarrow\mathbb{Z}$ is a quasi-polynomial if there exists a period $m$ and polynomials $f_0,\ldots,f_{m-1}$ such that $f(t)=f_{t\bmod m}(t)$ for all $t$. We conjecture that, if the $a_i(t)$ are polynomials (or quasi-polynomials) in $t$, then the Frobenius number agrees with a quasi-polynomial, for sufficiently large $t$. We prove this in the case where the $a_i(t)$ are linear functions, and also prove it in the case where $n$ (the number of generators) is at most 3.

scholarly journals Untrodden pathways in the theory of the restricted partition function p(n,N)

2021 ◽  
Vol 180 ◽  
pp. 105423
Author(s):  
Atul Dixit ◽  
Pramod Eyyunni ◽  
Bibekananda Maji ◽  
Garima Sood
Keyword(s):  
Partition Function ◽  
Restricted Partition

scholarly journals Solution of certain Pell equations

2018 ◽  
Vol 11 (04) ◽  
pp. 1850056 ◽  
Author(s):  
Zahid Raza ◽  
Hafsa Masood Malik
Keyword(s):  
Positive Integer ◽  
Fundamental Solution ◽  
Positive Solutions ◽  
Continued Fraction ◽  
Pell Equation ◽  
Lucas Sequences ◽  
Pell Equations ◽  
Integer Solutions ◽  
Positive Integers

Let [Formula: see text] be any positive integers such that [Formula: see text] and [Formula: see text] is a square free positive integer of the form [Formula: see text] where [Formula: see text] and [Formula: see text] The main focus of this paper is to find the fundamental solution of the equation [Formula: see text] with the help of the continued fraction of [Formula: see text] We also obtain all the positive solutions of the equations [Formula: see text] and [Formula: see text] by means of the Fibonacci and Lucas sequences.Furthermore, in this work, we derive some algebraic relations on the Pell form [Formula: see text] including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equation [Formula: see text] in terms of [Formula: see text] We extend all the results of the papers [3, 10, 27, 37].

scholarly journals On the restricted partition function

2018 ◽  
Vol 47 (3) ◽  
pp. 565-588 ◽  
Author(s):  
Mircea Cimpoeaş ◽  
Florin Nicolae
Keyword(s):  
Partition Function ◽  
Restricted Partition

scholarly journals New Results on Vector and Homing Vector Automata

2019 ◽  
Vol 30 (08) ◽  
pp. 1335-1361
Author(s):  
Özlem Salehi ◽  
Abuzer Yakaryılmaz ◽  
A. C. Cem Say
Keyword(s):  
Real Time ◽  
Nonnegative Integer ◽  
Diophantine Equations ◽  
Finite Automata ◽  
Separation Problem ◽  
Integer Solutions ◽  
Homing Vector

We present several new results and connections between various extensions of finite automata through the study of vector automata and homing vector automata. We show that homing vector automata outperform extended finite automata when both are defined over [Formula: see text] integer matrices. We study the string separation problem for vector automata and demonstrate that generalized finite automata with rational entries can separate any pair of strings using only two states. Investigating stateless homing vector automata, we prove that a language is recognized by stateless blind deterministic real-time version of finite automata with multiplication iff it is commutative and its Parikh image is the set of nonnegative integer solutions to a system of linear homogeneous Diophantine equations.

scholarly journals Nonnegative integer solutions of the equationFn

2019 ◽  
Vol 43 (3) ◽  
pp. 1115-1123 ◽  
Author(s):  
Fatih ERDUVAN ◽  
Refik KESKİN
Keyword(s):  
Nonnegative Integer ◽  
Integer Solutions
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