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

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