Refined upper bounds for the linear Diophantine problem of Frobenius

On a Linear Diophantine Problem of Frobenius

Let a1, a2, . . . , ak be positive and pairwise coprime integers with product P . For each i, 1 ≤ i ≤ k, set Ai = P/ai. We find closed form expressions for the functions g(A1, A2, . . . , Ak) and n(A1, A2, . . . , Ak) that denote the largest (respectively, the number of) N such that the equation A1x1 + A2x2 + · · · + Akxk = N has no solution in nonnegative integers xi. This is a special case of...

The Frobenius Coin Problem Upper Bounds on The Frobenius Number

In its simplest form, the coin problem is this: what is the largest positive amount of money that cannot be obtained using two coins of specified distinct denominations? For example, using coins of 2 units and 3 units, it is easy so see that every amount greater than or equal to 2 can be obtained, but 1 cannot be obtained. Using coins of 2 units and 5 units, every amount greater than or equal t...

The Diophantine Frobenius Problem

An Optimal Lower Bound for the Diophantine Problem of Frobenius

Given N positive integers a1, . . . , aN with GCD (a1, . . . , aN) = 1, let fN denote the largest natural number which is not a positive integer combination of a1, . . . , aN . This paper gives an optimal lower bound for fN in terms of the absolute inhomogeneous minimum of the standard (N − 1)-simplex.

The diophantine problem of Frobenius: A close bound

The conductor of n positive integer numbers a l,a2, ... ,an' whose greatest coII1mon divisor is equal'to I, is'defmed as the th.e minimal K, such that for every m ~K , the equation a1x 1+a2X 2+ ... +an Xn=m, h!ls a solution over the nOI}negative integers. In this notewe give a polYIlomial aigorithm computing'a close bound ~ for the conductor K o( n given positive integers, when n is fixed. The ...