Sets of Gaps in Sequences of Frobeniusʹ Problems with 3 Elements

Given a Frobeniusʹ Set A = { a 1 , . . . , a k − 1 , N } , with 1 ⩽ a 1 < … < a k − 1 < N and gcd ( a 1 , … a k − 1 , N ) = 1 , let us denote R ( A ) = { n ∈ N | ∃ λ 1 , … , λ k ∈ N : n = λ 1 a 1 + ⋯ + λ k − 1 a k − 1 + λ k N } and R ¯ ( A ) = N \ R ( A ) . The classical study of the Frobeniusʹ Problem for a given Frobeniusʹ set A is the computation of the number f ( A ) = max R ¯ ( A ) (also called the Frobeniusʹ Number) and the cardinal | R ¯ ( A ) | . It is a difficult task to explicitly compute the Solution Set R ¯ ( A ) , also known as the Set of Gaps. ork is focused on a tool for finding the Set of Gaps of a given sequence of Frobeniusʹ Sets when k = 3 .