On a Linear Diophantine Problem of Frobenius: Extending the Basis

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 Diophantine Frobenius Problem

on the effect of linear & non-linear texts on students comprehension and recalling

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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 ...

A geometric approach to the diophantine Frobenius problem

It turns out that all instances of the diophantine Frobenius problem for three coprime ai have a common geometric structure which is independent of arithmetic coincidences among the ai. By exploiting this structure we easily obtain Johnson’s formula for the largest non-representable z, as well as a formula for the number of such z. A procedure is described which computes these quantities in O(l...