Using Double-Loop digraphs for solving Frobeniusʹ Problems

Given a set A = { a 1 , … , a k } with 1 ⩽ a 1 < … < a k and gcd ( a 1 , … , a k ) = 1 , let us denote R ( A ) = { m ∈ N | ∃ x 1 , … , x k ∈ N : m = ∑ i = 1 k x 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 ) | . s work we propose a method to explicitly find the set R ¯ ( A ) in a closed form when k = 3 . As far as we know, this is the first proposed method to find a set R ¯ ( A ) .