Norms of indecomposable integers in real quadratic fields

Abstract

We study totally positive, additively indecomposable integers in a real quadratic field $\mathbb{Q}(\sqrt{D})$. We estimate the size of the norm of an indecomposable integer by expressing it as a power series in $u_i^{-1}$, where $\sqrt{D}$ has the periodic continued fraction expansion $[u_0,u_1,u_2,\dots ,u_{s-1},2u_0,u_1,u_2,\dots ]$. This enables us to disprove a conjecture of Jang-Kim [JK] concerning the maximal size of the norm of an indecomposable integer.

12 pages, to appear in J. Number Theory

References in corpus (2)

• Number fields without n-ary universal quadratic forms
• Universal quadratic forms and elements of small norm in real quadratic fields

Cited by in corpus (11)

• Lifting problem for universal quadratic forms
• Universal quadratic forms, small norms and traces in families of number fields
• There are no universal ternary quadratic forms over biquadratic fields
• Universal quadratic forms over multiquadratic fields
• Indecomposable integers in real quadratic fields
• Universal quadratic forms and indecomposables in number fields: A survey
• Additive structure of totally positive quadratic integers
• Distribution of class numbers in continued fraction families of real quadratic fields
• Trace and norm of indecomposable integers in cubic orders
• Additive structure of non-monogenic simplest cubic fields
• Arithmetic of cubic number fields: Jacobi-Perron, Pythagoras, and indecomposables