| Popis: |
We solve the 3-variable problem: find integers x ≥ 0, y ≥ 0, z ≥ 0 that satisfy ax + by + cz = L for given integers a, b, c, L, where 1 < a < b < c < L. The method of solution is related to the one for the Frobenius problem in three variables, which has been solved by Selmer and Beyer and by Rodseth (J. Reine Angew. Math.301 (1978), 161–178). These methods take O(a) steps, in the worst case, to find the Frobenius value. The method here, for the Frobenius value, is shown to be rapid, requiring less than O(log a) steps. The diophantine equation is then solved with little extra effort to result in an O(log a) method overall. |
