THE DIOPHANTINE FROBENIUS PROBLEM
CONTENTS ALGORITHMIC ASPECTS ALGORITHMS FOR COMPUTING G(A1, A2, A3) RØDSETH’S ALGORITHM DAVISON’S ALGORITHM KILLINGBERGTRؒS METHOD GENERAL ALGORITHMS SCARF AND SHALLCROSS’ METHOD HEAP AND LYNN METHOD GREENBERG’S ALGORITHM NIJENHUIS’ ALGORITHM WILF’S ALGORITHM KANNAN’S METHOD COMPUTATIONAL COMPLEXITY OF FP SUPPLEMENTARY NOTES THE FROBENIUS NUMBER FOR SMALL N COMPUTING G(P, Q) IS EASY A FORMULA FOR G(A1, A2, A3) RESULTS WHEN N = HOFMEISTER’S FORMULA AND ITS GENERALIZATION MORE SPECIAL CASES JOHNSON INTEGERS G(A1, A2, A3, A4) THE GENERAL PROBLEM FORMULAS AND UPPER BOUNDS BOUNDS IN TERMS OF THE LCM(A1, , AN) ARITHMETIC AND RELATED SEQUENCES REGULAR BASES EXTENDING BASIS LOWER BOUNDS SYLVESTER DENUMERANT FROM PARTITIONS TO DENUMERANTS FORMULAS AND BOUNDS FOR D(M; A1, , AN) COMPUTING DENUMERANTS PARTIAL FRACTIONS BELL’S METHOD D(M; P, Q) D(M; A1, A2, A3) AND D(M; A1, A2, A3, A4) HILBERT SERIES A PROOF OF A FORMULA FOR G(A1, A2, A3) EHRHART POLYNOMIAL VARIATIONS OF THE DENUMERANT D?(M; A1, , AN) D??(M; A1, , AN) 4 10 SUPPLEMETARY NOTES INTEGERS WITHOUT REPRESENTATION SYLVESTER’S CLASSICAL RESULT NIJENHUIS’ AND WILF’S RESULTS FORMULAS FOR N(A1, , AN) ARITHMETIC SEQUENCES THE SUM OF INTEGERS IN N(P, Q) RELATED GAMES SYLVER COINAGE THE JUGS PROBLEM SUPPLEMETARY NOTES GENERALIZATIONS AND RELATED PROBLEMS SPECIAL FUNCTIONS THE MODULAR GENERALIZATION THE POSTAGE STAMP PROBLEM (A1, , AN)-TREES VECTOR GENERALIZATION OF FP NUMERICAL SEMIGROUPS GAPS AND NON-GAPS TELESCOPIC SEMIGROUPS HYPERELLIPTIC SEMIGROUPS SYMMETRIC SEMIGROUPS INTERSECTION OF SEMIGROUPS AP´ERY SETS RELATED CONCEPTS TYPE SEQUENCES COMPLETE INTERSECTION THE M¨OBIUS FUNCTION APPLICATIONS OF THE FROBENIUS NUMBER COMPLEXITY ANALYSIS OF THE SHELL-SORT METHOD PETRI NETS P/T SYSTEMS WEIGHTED CIRCUITS SYSTEMS PARTITION OF A VECTOR SPACE MONOMIAL CURVES ALGEBRAIC GEOMETRIC CODES TILINGS APPLICATIONS OF DENUMERANTS BALLS AND CELLS CONJUGATE POWER EQUATIONS INVARIANT CUBATURE FORMULAS OTHER APPLICATIONS GENERATING RANDOM VECTORS NON-HAMILTONIAN GRAPHS