相关论文: Frobenius numbers associated with Diophantine trip…

For $ k \geq 2 $, we let $ A = (a_{1}, a_{2}, \ldots, a_{k}) $ be a $k$-tuple of positive integers with $\gcd(a_{1}, a_2, \ldots, a_k) =1$ and, for a non-negative integer $s$, the generalized Frobenius number of $A$, $g(A;s) = g(a_1, a_2,…

The Frobenius coin problem in three variables, for three positive relatively prime integers $a_1< a_2< a_3$ asks to find the largest number not representable as $a_1x_1+a_2x_2+a_3x_3$ with non-negative integer coefficients $x_1$, $x_2$ and…

Given relatively prime positive integers, $a_1,\ldots,a_n$, the Frobenius number is the largest integer with no representations of the form $a_1x_1+\cdots+a_nx_n$ with nonnegative integers $x_i$. This classical value has recently been…

For given positive integers $a_1,a_2,\dots,a_k$ with $\gcd(a_1,a_2,\dots,a_k)=1$, the denumerant $d(n)=d(n;a_1,a_2,\dots,a_k)$ is the number of nonnegative solutions $(x_1,x_2,\dots,x_k)$ of the linear equation $a_1 x_1+a_2 x_2+\dots+a_k…

In this work, we prove the following result(Theorem 1): Suppose that n is a positive integer, p an odd prime, and such that either n is congruent to 0 modulo4 and p congruent to 3 modulo8; or alternatively, n is congruent to 2 modulo4 and p…

The objective of the paper is to determine the complete solutions for the Diophantine equation $x^2 + 3^{\alpha}113^{\beta} = y^{\mathfrak{n}}$ in positive integers $x$ and $y$ (where $x, y \geq 1$), non-negative exponents $\alpha$ and…

A number $\alpha$ has a representation with respect to the numbers $\alpha_1,...,\alpha_n$, if there exist the non-negative integers $\lambda_1,... ,\lambda_n$ such that $\alpha=\lambda_1\alpha_1+...+\lambda_n \alpha_n$. The largest natural…

The Frobenius number for a set of relatively prime positive integers, where the smallest integer in the set is at least 2, is the largest integer that cannot be expressed as a nonnegative linear combination of those integers. We analyze the…

In this note, we find all the solutions of the Diophantine equation x^2 +2^a.3^b.11^c=y^n in nonnegative integers a, b, c, x, y, n>= 3 with x and y coprime.

Let $p$ be a prime integer, $\mathbb{Z}_p$ the finite field of order $p$ and $\mathbb{Z}^{*}_{p}$ is its multiplicative cyclic group. We consider the Diophantine equation $x^n + y^n = z^n$ with $1 \leq n \leq \frac{p - 1}{2}$. Our main aim…

Let $N \geq2$ and let $1 < a_1 < ... < a_N$ be relatively prime integers. Frobenius number of this $N$-tuple is defined to be the largest positive integer that cannot be expressed as $\sum_{i=1}^N a_i x_i$ where $x_1,...,x_N$ are…

For given coprime positive integers $a$ and $b$, the classical Frobenius coin problem asked to find the largest number that cannot be expressed in the form $ax+by$ for nonnegative integers $x$ and $y$, also known as the Frobenius number.…

Generalizing an argument of Matiyasevich, we illustrate a method to generate infinitely many diophantine equations whose solutions can be completely described by linear recurrences. In particular, we provide an integer-coefficient…

We consider a generalization of the Frobenius Problem where the object of interest is the greatest integer which has exactly $j$ representations by a collection of positive relatively prime integers. We prove an analogue of a theorem of…

We study the exponential Diophantine equation $x^2+p^mq^n=2y^p$ in positive integers $x,y,m,n$, and odd primes $p$ and $q$ using primitive divisors of Lehmer sequences in combination with elementary number theory. We discuss the solvability…

Let $k\ge 2$ and $a_1, a_2, \cdots, a_k$ be positive integers with \[ \gcd(a_1, a_2, \cdots, a_k)=1. \] It is proved that there exists a positive integer $G_{a_1, a_2, \cdots, a_k}$ such that every integer $n$ strictly greater than it can…

Let $n$ be a positive integer. We discuss pairs of distinct odd primes $p$ and $q$ not dividing $n$ for which the Diophantine equations $pq=x^2+ny^2$ have integer solutions in $x$ and $y$. As its examples we classify all such pairs of $p$…

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 solve the Frobenius problem for the numerical…

For $ k \geq 2 $, let $ A = (a_{1}, a_{2}, \ldots, a_{k}) $ be a $k$-tuple of positive integers with $\gcd(a_{1}, a_2, \ldots, a_k) = 1$. For a non-negative integer $s$, the generalized Frobenius number of $A$, denoted as $\mathtt{g}(A;s) =…

Let $F_n$ denote the $n^{th}$ term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation $F_1^p+2F_2^p+\cdots+kF_{k}^p=F_{n}^q$ in the positive integers $k$ and $n$, where $p$ and $q$ are given positive integers.…