Explore the vast range of titles available.

We give an explicit formula for the p-Frobenius number of triples associated with Diophantine Equations x[sup.2]−y[sup.2]=z[sup.r] (r≥2), that is, the largest positive integer that can only be represented in p ways by combining the three integers of the solutions of Diophantine equations x[sup.2]−y[sup.2]=z[sup.r]. This result is also a generalization because if r=2 and p=0, the (0-)Frobenius number of the Pythagorean triple has already been given. To find p-Frobenius numbers, we use geometrically easy to understand figures of the elements of the p-Apéry set, which exhibits symmetric appearances.

The famous linear diophantine problem of Frobenius is the problem to determine the largest integer (Frobenius number) whose number of representations in terms of a 1 , ⋯ , a k is at most zero, that is not representable. In other words, all the integers greater than this number can be represented for at least one way. One of the natural generalizations of this problem is to find the largest integer (generalized Frobenius number) whose number of representations is at most a given nonnegative integer p . It is easy to find the explicit form of this number in the case of two variables. However, no explicit form has been known even in any special case of three variables. In this paper we are successful to show explicit forms of the generalized Frobenius numbers of the triples of triangular numbers. When p = 0 , their Frobenius number is given by Robles-Pérez and Rosales in 2018.

In this paper, we introduce and study the numerical semigroups generated by {a1, a2, . . .} ⊂ N such that a1 is the repunit number in base b > 1 of length n > 1 and ai − ai−1 = a bi−2, for every i ≥ 2, where a is a positive integer relatively prime with a1. These numerical semigroups generalize the repunit numerical semigroups among many others. We show that they have interesting properties such as being homogeneous and Wilf. Moreover, we solve the Frobenius problem for this family, by giving a closed formula for the Frobenius number in terms of a, b and n, and compute other usual invariants such as the Ap´ery sets, the genus or the type.

In this paper, we give closed-form expressions of the p-Frobenius number for the triple of the generalized star numbers an(n−1)+1 for an integer a≥4. When a=6, it is reduced to the famous star number. For the set of given positive integers a1,a2,…,ak, the p-Frobenius number is the largest integer N whose number of non-negative integer representations N=a1x1+a2x2+⋯+akxk is at most p. When p=0, the 0-Frobenius number is the classical Frobenius number, which is the central topic of the famous linear Diophantine problem of Frobenius.

For given positive integers a1,a2,⋯,ak with gcd(a1,a2,⋯,ak)=1, the denumerant d(n)=d(n;a1,a2,⋯,ak) is the number of nonnegative solutions (x1,x2,⋯,xk) of the linear equation a1x1+a2x2+⋯+akxk=n for a positive integer n. For a given nonnegative integer p, let Sp=Sp(a1,a2,⋯,ak) be the set of all nonnegative integer n’s such that d(n)>p. In this paper, by introducing the p-numerical semigroup, where the set N0 is finite, we give explicit formulas of the p-Frobenius number, which is the maximum of the set N0 and related values for the triple of arithmetic progressions. The main aim is to determine the elements of the p-Apéry set.

We give an explicit formula for the p-Frobenius number of triples associated with Diophantine Equations x2−y2=zr (r≥2), that is, the largest positive integer that can only be represented in p ways by combining the three integers of the solutions of Diophantine equations x2−y2=zr. This result is also a generalization because if r=2 and p=0, the (0-)Frobenius number of the Pythagorean triple has already been given. To find p-Frobenius numbers, we use geometrically easy to understand figures of the elements of the p-Apéry set, which exhibits symmetric appearances.

Many people, including Horadam, have studied the numbers Wn, satisfying the recurrence relation Wn=uWn−1+vWn−2 (n≥2) with W0=0 and W1=1. In this paper, we study the p-numerical semigroups of the triple (Wi,Wi+2,Wi+k) for integers i,k(≥3). For a nonnegative integer p, the p-numerical semigroup Sp is defined as the set of integers whose nonnegative integral linear combinations of given positive integers a1,a2,…,aκ with gcd(a1,a2,…,aκ)=1 are expressed in more than p ways. When p=0, S=S0 is the original numerical semigroup. The largest element and the cardinality of N0∖Sp are called the p-Frobenius number and the p-genus, respectively.

In this paper, we introduce and study the numerical semigroups generated by { a 1 , a 2 , … } ⊂ N such that a 1 is the repunit number in base b > 1 of length n > 1 and a i - a i - 1 = a b i - 2 , for every i ≥ 2 , where a is a positive integer relatively prime with a 1 . These numerical semigroups generalize the repunit numerical semigroups among many others. We show that they have interesting properties such as being homogeneous and Wilf. Moreover, we solve the Frobenius problem for this family, by giving a closed formula for the Frobenius number in terms of a ,  b and n , and compute other usual invariants such as the Apéry sets, the genus or the type.

Let$ S $be a given finite set of positive and relatively prime integers. Denote$ L(S) $to be the set of integers obtained by taking all nonnegative integer linear combinations of integers in$ S $ . It is well known that there are finitely many positive integers that are not in$ L(S) $ . Let$ g(S) $and$ n(S) $represent the greatest integer that does not belong to$ L(S) $and the number of nonnegative integers that do not belong to$ L(S) $ , respectively. The Frobenius problem is to determine$ g(S) $and$ n(S) $ . In 2016, Tripathi obtained results on$ g(S) $and$ n(S) $when$ S = \\{a, ha+d, ha+db, ha+db^2, \\ldots, ha+db^k\\} $ . In this paper, for$ S_c: = \\{a, ha+d, ha+c+db, ha+2c+db^2, \\ldots, ha+kc+db^k\\} $with$ h, c $being nonnegative integers,$ a, b, d $being positive integers and$ \\gcd(a, d) = 1 $ , we focused the investigation on formulas for$ g(S_c) $and$ n(S_c) $ . Actually, we gave formulas for$ g(S_c) $and$ n(S_c) $for all sufficiently large values of$ d $when$ c $is any multiple of$ d $or certain multiples of$ a $ . This generalized the results of Tripathi in 2016.

In this paper, we study the (classical) Frobenius problem, namely the problem of finding the largest integer that cannot be represented as a nonnegative integer combination of given, relatively prime, (strictly) positive integers (known as the Frobenius number). The main contribution of this paper are observations regarding a previously known upper bound on the Frobenius number where, in particular, we observe that a previously presented argument features a subtle error, which alters the value of the upper bound. Despite this, we demonstrate that the subtle error does not impact upon on the validity of the upper bound, although it does impact on the upper bounds tightness. Notably, we formally state the corrected result and additionally compare the relative tightness of the corrected upper bound with the original. In particular, we show that the updated bound is tighter in all but only a relatively “small” number of cases using both formal techniques and via Monte Carlo simulation techniques.