Some contributions to the Frobeniusʹ Problem

Given a set A = { a 1 , … , a k } with 1 ⩽ a 1 < … < a k and g cd ( a 1 , … , a k ) = 1 , let us denote R ( A ) = { m ∈ N | ∃ λ 1 , … , λ k ∈ N : m = ∑ i = 1 k λ i a i } and R ¯ ( A ) = N \ R ( A ) . The classical study of the Frobeniusʹ Problem for a given set A is the computation of the number f ( A ) = max R ¯ ( A ) (also called the Frobenius Number) and | R ¯ ( A ) | . In this work we propose a method to explicitly find the set | R ¯ ( A ) . We treat the case k = 3 in a different way from the case k > 3 .