Abstract:
The problem of finding the Frobenius number is urgent, it is closely related to the graph theory and the well-known knapsack and exchange problems. The formula for calculating the Frobenius number is known only for two numbers: if the greatest common divisor of natural numbers $a$ and $b$ is one, then $\mathrm{frob}(a,b) = ab - a - b$. The paper proves two properties on the problem of finding the Frobenius number for three natural numbers. The first of these properties complements, even enriches, the mentioned famous Frobenius formula for two natural numbers. This property allows moving from a Frobenius number for two natural numbers to a Frobenius number for three natural numbers when the third number is a Frobenius number for the first two numbers: $\mathrm{frob}(a,b,\mathrm{frob}(a,b)) = \mathrm{frob} (a,b) - a$. The second property belongs to the “homogeneity” properties of the frob operation, allowing in some cases to take the multiplier before one or more variables beyond the sign of the operation. As a result of this property, the problem of finding the Frobenius number for three numbers of special form is reduced to the well-known formula for the Frobenius number for three consecutive natural numbers.
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