Bachmann studied mathematics at the University of Berlin and received his doctorate in 1862 for his thesis on group theory. He then went to Breslau to study for his habilitation, which he received in 1864 for his thesis on complex unit s.
His major works include
- Analytische Zahlentheorie, a work on analytic number theory in which Big O notation was first introduced
- Niedere Zahlentheorie, a two-volume work on elementary number theory
- Das Fermat-Problem in seiner bisherigen Entwicklung, a work about Fermat's Last Theorem