Totally positive elements with m partitions exist in almost all real quadratic fields

In this paper, we study partitions of totally positive integral elements α in a real quadratic field K . We prove that for a fixed integer m ≥ 1 , an element with m partition exists in almost all K . We also obtain an upper bound for the norm of α that can be represented as a sum of indecomposables in at most m ways, completely characterize the α ’s represented in exactly $2$ ways, and subsequently apply this result to complete the search for fields containing an element with m partitions for 1 ≤ m ≤ 7 .