On totient abundant numbers.
We compute the numbers of locally principal ideals with given norm in a class of definite quaternion orders and the traces of the Brandt-Eichler matrices corresponding to these orders. As an application, we compute the numbers of representations of algebraic integers by the norm forms of definite quaternion orders with class number one as well as we obtain class number relations for some CM-fields.
We prove two supercongruences involving Almkvist-Zudilin sequences, which were originally conjectured by Z.-H. Sun (2020).