1. Frobenius Number for Three Numbers By Arash Farahmand MATH 870 Spring 2007

2. Coin Exchange Problem What is the largest amount that cannot be changed? What is the largest amount that cannot be changed? First tackled by G. Frobenius (1849-1917) and J. Sylvester (1814-1897) First tackled by G. Frobenius (1849-1917) and J. Sylvester (1814-1897)

3. N = 2 With two coins With two coins Formula: g (n, m) = nm – n – m Formula: g (n, m) = nm – n – m

4. Complexity for N > 2 Frobenius numbers by lattice point enumeration Frobenius numbers by lattice point enumeration Polynomial time on average for fixed N Polynomial time on average for fixed N Relatively fast algorithm for lattice reduction applied to find Frobenius number when N = 3 Relatively fast algorithm for lattice reduction applied to find Frobenius number when N = 3

5. Algorithm Step 1. Form the homogeneous basis and then use lattice reduction to obtain a reduced basis V Step 1. Form the homogeneous basis and then use lattice reduction to obtain a reduced basis V Step 2. Make use of V and the ILP methods to determine the two axial protoelbows (x 1, y 1 ) and (x 2, y 2 ) Step 2. Make use of V and the ILP methods to determine the two axial protoelbows (x 1, y 1 ) and (x 2, y 2 )

6. Algorithm (continued) Step 3. If y 1 or x 2 is zero then the elbow set is {(x 1, 0), (0, y 2 )}; otherwise it is {(x 1, 0), (0, y 2 ), (x 1 + x 2, y 1 + y 2 )} Step 3. If y 1 or x 2 is zero then the elbow set is {(x 1, 0), (0, y 2 )}; otherwise it is {(x 1, 0), (0, y 2 ), (x 1 + x 2, y 1 + y 2 )} Step 4. In all cases the Frobenius number is max [{(x 1, y 1 + y 2 ), (x 1 + x 2, y 2 } B] - ΣA Step 4. In all cases the Frobenius number is max [{(x 1, y 1 + y 2 ), (x 1 + x 2, y 2 } B] - ΣA

7. References M. Beck and S. Robins, Computing the Continuous Discretely, 2006, Springer M. Beck and S. Robins, Computing the Continuous Discretely, 2006, Springer S. Wagon, D. Einstein, D. Lichtblau, and A. Strzenbonski, Frobenius Numbers by Lattice Point Enumeration, http://stanwagon.com/public/FrobeniusByLatt icePoints.pdf, Revised Aug 1, 2006, last visited February 15, 2007 S. Wagon, D. Einstein, D. Lichtblau, and A. Strzenbonski, Frobenius Numbers by Lattice Point Enumeration, http://stanwagon.com/public/FrobeniusByLatt icePoints.pdf, Revised Aug 1, 2006, last visited February 15, 2007 http://stanwagon.com/public/FrobeniusByLatt icePoints.pdf http://stanwagon.com/public/FrobeniusByLatt icePoints.pdf