Totally positive elements with <i>m</i> partitions exist in almost all real quadratic fields

Abstract 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 \geq 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 \leq m \leq 7 $ .