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For positive and relatively prime set of integers A, let Γ ( A ) denote the set of integers that is formed by taking nonnegative integer linear combinations of integers in A. Then there are finitely many positive integers that do not belong to Γ ( A ) . For A = { a , h a + d , h a + b d , h a + b 2 d , … , h a + b k d } , gcd ⁡ ( a , d ) = 1 , we determine the largest integer g ( A ) that does not belong to Γ ( A ) , and the number of integers n ( A ) that does not belong to Γ ( A ) , both for all sufficiently large values of d. This extends a result of Selmer, and corrects a result of Hofmeister, both given in special cases.

Let n be a positive integer and T ( n ) the numerical semigroup generated by { 3.2 n + i − 1 | i ∈ N } . In this paper, we give formulas for the Frobenius number, the gender, the embedding dimension and the type of T ( n ) .

Given N ⩾ 2 positive integers a 1 , a 2 , … , a N with GCD ( a 1 , … , a N ) = 1 , let f N denote the largest natural number which is not a positive integer combination of a 1 , … , a N . This paper gives an optimal lower bound for f N in terms of the absolute inhomogeneous minimum of the standard ( N − 1 ) -simplex.

In the Frobenius problem with two variables, one is given two positive integers a and b that are relative prime, and is concerned with the set of positive numbers NR that have no representation by the linear form ax + by in nonnegative integers x and y. We give a complete characterization of the set NR, and use it to establish a relation between the power sums over its elements and the power sums over the natural numbers. This relation is used to derive new recurrences for the Bernoulli numbers.

We study the number of lattice points in integer dilates of the rational polytope P={(x1,…,xn)∈R⩾0n:∑k=1nxkak⩽1}, where a1,…,an are positive integers. This polytope is closely related to the linear Diophantine problem of Frobenius: given relatively prime positive integers a1,…,an, find the largest value of t (the Frobenius number) such that m1a1+···+mnan=t has no solution in positive integers m1,…,mn. This is equivalent to the problem of finding the largest dilate tP such that the facet {∑k=1nxkak=t} contains no lattice point. We present two methods for computing the Ehrhart quasipolynomials L(P,t)≔#(tP∩Zn) and L(P°,t)≔#(tP°∩Zn). 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.

In this paper, we characterize those numerical semigroups containing 〈 n 1 , n 2 〉 . From this characterization, we give formulas for the genus and the Frobenius number of a numerical semigroup. These results can be used to give a method for computing the genus and the Frobenius number of a numerical semigroup with embedding dimension three in terms of its minimal system of generators.

Let A ⊆[0; l ] be a set of n integers, and let h ⩾2. By how much does | hA | exceed |( h −1) A | ? How can one estimate | hA | in terms of n , l ? We give sharp lower bounds extending and generalizing the well-known theorem of Freiman for |2 A |. A number of applications are provided as well. In particular, we give a solution for the old extremal problem of Frobenius–Erdos–Graham concerning estimating of the largest integer, non-representable by a linear form. In a sense, our solution can not be improved.