| Let a(,1),...,a(,k) be integers having no common divisor exceeding one. The question of determining the largest integer g(,k) = g(a(,1),...,a(,k)) that is not representable in the form.;with the x(,i) nonnegative integers was proposed by G. Frobenius in the nineteenth century.;For k = 2 the problem has been solved by J. J. Sylvester {43} in 1884, who showed.;g(a(,1),a(,2)) = (a(,1)-1)(a(,2)-1)-1.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;Since that time some progress has been made in establishing algorithms and bounds for g(,k) by a variety of researchers. However no general explicit formula even for g(,3) has been conjectured. Four special cases for g(,3) in which the explicit formulas are known are g(t,t + 1,t + 2),g(t,t + 1,t + z) for z > 2 (due to J. B. Roberts), g(a(,1),a(,2),a(,3)) where a(,2) (TBOND) -a(,3) (mod a(,1)) with a(,1),a(,2),a(,3) relatively prime in pairs (due to A. Brauer and J. E. Shockley) and g(t,t + y,t + yz) where z (GREATERTHEQ) 0 and y an integer (due to G. R. Hofmeister).;The present work develops an algorithm for g(,3) which leads to the following explicit formula for g(,3): For 0 < a < b < c with gcd(a,b,c) = 1, d = gcd(c-b,c-a),;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;defined by.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;provided.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;The constructive proof of this formula based on an algorithm is given. It will be shown that the already known special cases mentioned above follow from this. Finally some further problems are proposed. |
