The Frobenius Problem and Its Generalizations

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Let x 1, x 2, ..., x n be positive integers. It is well-known that every sufficiently large integer can be represented as a non-negative integer linear combination of the x i if and only if \(\gcd(x_1, x_2, \ldots, x_n) = 1\). The Frobenius problem is the following: given positive integers x 1, x 2, ..., x n with \(\gcd(x_1, x_2, \ldots, x_n) = 1 \), compute the largest integer not representable as a non-negative integer linear combination of the x i . This largest integer is sometimes denoted g(x 1,..., x n ).KeywordsRegular ExpressionLarge IntegerRegular LanguageLonge WordFrobenius NumberThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.