Linear Diophantine Problem Research Articles

P-Numerical semigroups with p-symmetric properties

The so-called Frobenius number in the famous linear Diophantine problem of Frobenius is the largest integer such that the linear equation [Formula: see text] ([Formula: see text] are given positive integers with [Formula: see text]) does not have a non-negative integer solution [Formula: see text]. The generalized Frobenius number (called the [Formula: see text]-Frobenius number) is the largest integer such that this linear equation has at most [Formula: see text] solutions. That is, when [Formula: see text], the [Formula: see text]-Frobenius number is the original Frobenius number. In this paper, we introduce and discuss [Formula: see text]-numerical semigroups by developing a generalization of the theory of numerical semigroups based on this flow of the number of representations. That is, for a certain non-negative integer [Formula: see text], [Formula: see text]-gaps, [Formula: see text]-symmetric semigroups, [Formula: see text]-pseudo-symmetric semigroups, and the like are defined, and their properties are obtained. When [Formula: see text], they correspond to the original gaps, symmetric semigroups and pseudo-symmetric semigroups, respectively.

The Frobenius Number for Jacobsthal Triples Associated with Number of Solutions

In this paper, we find a formula for the largest integer (p-Frobenius number) such that a linear equation of non-negative integer coefficients composed of a Jacobsthal triplet has at most p representations. For p=0, the problem is reduced to the famous linear Diophantine problem of Frobenius, the largest integer of which is called the Frobenius number. We also give a closed formula for the number of non-negative integers (p-genus), such that linear equations have at most p representations. Extensions to the Jacobsthal polynomial and the Jacobsthal–Lucas polynomial give more general formulas that include the familiar Fibonacci and Lucas numbers. A basic problem with the Fibonacci triplet was dealt by Marin, Ramírez Alfonsín and M. P. Revuelta for p=0 and by Komatsu and Ying for the general non-negative integer p.

The Frobenius Number for Sequences of Triangular Numbers Associated with Number of Solutions

The famous linear diophantine problem of Frobenius is the problem to determine the largest integer (Frobenius number) whose number of representations in terms of $a_1,\dots,a_k$ 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-P\'erez and Rosales in 2018.

The p-Frobenius and p-Sylvester numbers for Fibonacci and Lucas triplets.

In this paper we study a certain kind of generalized linear Diophantine problem of Frobenius. Let $ a_1, a_2, \dots, a_l $ be positive integers such that their greatest common divisor is one. For a nonnegative integer $ p $, denote the $ p $-Frobenius number by $ g_p (a_1, a_2, \dots, a_l) $, which is the largest integer that can be represented at most $ p $ ways by a linear combination with nonnegative integer coefficients of $ a_1, a_2, \dots, a_l $. When $ p = 0 $, the $ 0 $-Frobenius number is the classical Frobenius number. When $ l = 2 $, the $ p $-Frobenius number is explicitly given. However, when $ l = 3 $ and even larger, even in special cases, it is not easy to give the Frobenius number explicitly. It is even more difficult when $ p > 0 $, and no specific example has been known. However, very recently, we have succeeded in giving explicit formulas for the case where the sequence is of triangular numbers [1] or of repunits [2] for the case where $ l = 3 $. In this paper, we show the explicit formula for the Fibonacci triple when $ p > 0 $. In addition, we give an explicit formula for the $ p $-Sylvester number, that is, the total number of nonnegative integers that can be represented in at most $ p $ ways. Furthermore, explicit formulas are shown concerning the Lucas triple.

On polynomial-time solvable linear Diophantine problems

We obtain a polynomial-time algorithm that, given input (A, b), where A=(B|N) is an integer mxn matrix, m<n, with nonsingular mxm submatrix B and b is an m-dimensional integer vector, finds a nonnegative integer solution to the system Ax=b or determines that no such solution exists, provided that b is located sufficiently "deep" in the cone generated by the columns of B. This result improves on some of the previously known conditions that guarantee polynomial-time solvability of linear Diophantine problems.

Indeterminate linear problems from Asia to Europe

In the Chinese tradition, linear Diophantine problems can be traced back to two main categories. The first group includes the problems consisting of 2-equation systems with integer coefficients in n unknowns with $$n> 2$$ and integer solutions, such as: $$x_1 + x_2 + \dots + x_n = p$$ , $$b_1 x_1 + b_2 x_2+ \dots + b_n x_n = q$$ . The so-called “100 fowls problem” belongs to this group. The oldest statement can apparently be found in the Zhang Qiujian Suanjing (The Computation Classic of Zhang Qiujian), towards the second half of the 5th century AD (468–486). In the second category there are problems that can be represented through simultaneous linear congruences, which are generally formulated as follows: find a number x that, divided by $$m_1$$ , $$m_2$$ , $$m_3, \ldots , m_i$$ , give as remainders $$r_1$$ , $$r_2$$ , $$r_3, \ldots , r_i$$ . This kind of problem is currently called “Chinese (remainder) problem”. The oldest formulation is attributed to Master Sun Tzu (between 280 and 473). We shall present the different statements of the two problems in their circulation from Asia to Europe and the main proof procedures.

Graphic sequences of trees and a problem of Frobenius

We give a necessary and sufficient condition for the existence of a tree of order n with a given degree set. We relate this to a well-known linear Diophantine problem of Frobenius.

Refined upper bounds for the linear Diophantine problem of Frobenius

We study the Frobenius problem: given relatively prime positive integers a 1,…, a d , find the largest value of t (the Frobenius number g( a 1,…, a d )) such that ∑ k=1 d m k a k = t has no solution in nonnegative integers m 1,…, m d . We introduce a method to compute upper bounds for g( a 1, a 2, a 3), which seem to grow considerably slower than previously known bounds. Our computations are based on a formula for the restricted partition function, which involves Dedekind–Rademacher sums, and the reciprocity law for these sums.

The Frobenius Problem, Rational Polytopes, and Fourier–Dedekind Sums

We study the number of lattice points in integer dilates of the rational polytope P={(x 1,…,x n)∈ R ⩾0 n: ∑ k=1 n x ka k⩽1} , where a 1,…, a n are positive integers. This polytope is closely related to the linear Diophantine problem of Frobenius: given relatively prime positive integers a 1,…, a n , find the largest value of t (the Frobenius number) such that m 1 a 1+···+ m na n = t has no solution in positive integers m 1,…, m n . This is equivalent to the problem of finding the largest dilate t P such that the facet {∑ k=1 n x k a k = t} contains no lattice point. We present two methods for computing the Ehrhart quasipolynomials L( P , t)≔#( t P ∩ Z n ) and L( P °, t)≔#( t P °∩ Z n ). Within the computations a Dedekind-like finite Fourier sum appears. We obtain a reciprocity law for these sums, generalizing a theorem of Gessel. As a corollary of our formulas, we rederive the reciprocity law for Zagier's higher-dimensional Dedekind sums. Finally, we find bounds for the Fourier–Dedekind sums and use them to give new bounds for the Frobenius number.

The Frobenius Problem, Rational Polytopes, and Fourier–Dedekind Sums,

We study the number of lattice points in integer dilates of the rational polytope P ={(x 1 ,…,x n )∈ R ⩾0 n : ∑ k=1 n x k a k ⩽1} , where a 1 ,…, a n are positive integers. This polytope is closely related to the linear Diophantine problem of Frobenius : given relatively prime positive integers a 1 ,…, a n , find the largest value of t (the Frobenius number ) such that m 1 a 1 +···+ m n a n = t has no solution in positive integers m 1 ,…, m n . This is equivalent to the problem of finding the largest dilate t P such that the facet {∑ k =1 n x k a k = t } contains no lattice point. We present two methods for computing the Ehrhart quasipolynomials L ( P , t )≔#( t P ∩ Z n ) and L ( P °, t )≔#( t P °∩ Z n ). Within the computations a Dedekind-like finite Fourier sum appears. We obtain a reciprocity law for these sums, generalizing a theorem of Gessel. As a corollary of our formulas, we rederive the reciprocity law for Zagier's higher-dimensional Dedekind sums. Finally, we find bounds for the Fourier–Dedekind sums and use them to give new bounds for the Frobenius number.

Gradient elements of the knapsack polytope

The knapsack problem is a classical NP-hard problem of special interest in combinatorial mathematics and complexity theory. Particularly interesting are the properties of the knapsack problem domain. In this paper we study the set of gradient elements of the knapsack polytope. They are related with the well-known greedy algorithm for finding approximate solutions to the optimization knapsack problem and constitute an important subset of the extremal elements (vertices) of the knapsack polytope. We obtain a tight bound for the number of distinct gradient elements, which is an exponential function of the problem dimension. We also use the greedy-algorithmic approach and the concept of gradient elements to devise a polynomial algorithm for a large subclass of a linear Diophantine problem which is a variant of the knapsack problem. Some optimization issues related to the problems considered are discussed as well.

On consecutive subset sums

We consider the subset sums analog of the linear Diophantine problem of Frobenius. It is shown that if A ⊑ [1; l] is a sufficiently dense set of n positive integers, then [2 l − 2n + 1; σ − (2 l − 2 n + 1)] ⊑ A *, where σ is the sum of all elements of A, and A * is the set of all subset sums of A. The interval above is best possible and cannot be extended.

The linear diophantine problem of Frobenius for subsets of arithmetic sequences

Let A k = {a 1,. . . , a k } $ \subset \Bbb N $ with gcd (a 1,. . . , a k ) = 1. We shall say that a natural number n has a representation by a 1,. . . , a k if $ n =\sum \limits_{i=1}^{k}a_ix_i,\; x_i\in \Bbb N_0 $ . Let g = g (A k ) be the largest integer with no such representation. We then study the set A k = {a,ha + d,ha + 2d,..., ha + (k - 1) d} h,d > 0, gcd (a,d) = 1). If l k denotes the greatest number of elements which can be omitted without altering g (A k ), we show that ¶¶ $ 1-{4 \over \sqrt k} \le {l_k\over k} \le 1 - {3\over k}, $ ¶¶ provided a > k, or a = k with $ d \ge 2 h \sqrt {k} $ . The lower bound can be improved to 1 - 4 / k if we choose a > (k - 4) k + 3. Moreover, we determine sets $ E_k \subset A_k $ such that g (A k \ E k ) = g (A k ).

Optimal Representations by Sumsets and Subset Sums

LetAbe a set of non-negative integers. In our previous paper forh∈Z,h⩾2 we estimated from below the cardinality ofhAand this allowed us to obtain sharp estimates for suchG, that every integerg⩾Gmay be represented by a sum of elements ofA(the linear diophantine problem of Frobenius). However, “not only the number of elements ofhAis important, but also the way they are situated” (G. Freiman). In this paper we show thathAalways contains long continuous chains of integers and derive an estimation for the number of summands required for representation ofgof the above mentioned type.

Linear diophantine problems

Linear diophantine problems

On a linear diophantine problem of Frobenius

In this paper, linear diophantine problem of Frobenius is discussed. A theorem concerning the largest integer g m (a 1,a 2) and the smallest integer G m (a 1,a 2) with m different representations with a 1,a 2 as basis is proved.

The complexity of almost linear diophantine problems

We studied formulas of elementary number theory resulting from formulas of Presburger arithmetic PrA (additive elementary theory of integers with order) by substituting for some variables, polynomials and integer values of rational functions in a single new variable y, and quantifying over y. We show that the extension ALA (almost linear arithmetic) of PrA obtained in this way, has essentially the same upper and lower complexity bounds as the original theory. The same applies to the fragments of ALA obtained by restricting the number or type of quantifiers in formulas. We also show new upper complexity bounds for quantifier elimination in PrA and its fragments. The results form a common extension of known complexity bounds for PrA and for the existential almost linear problems studied by Gurari & Ibarra. The method is applicable also to other extensions of PrA without order, related to the famous bounds for binary forms of A. Baker.

On a linear diophantine problem for geometrical type sequences

Etude d'un probleme diophantien lineaire pour les suites de type geometrique et generalisation d'un resultat de E. L. Goldberg

On the linear diophantine problem of Frobenius in three variables.

On the linear diophantine problem of Frobenius in three variables.

Bounds for a Linear Diophantine Problem of Frobenius

Bounds for a Linear Diophantine Problem of Frobenius