**Paul Bachmann** was a German mathematician best known for his work in analytic number theory and for introducing the O-notation.

- It was undoubtedly his skills which turned Bachmann into a top class mathematician.
- Bachmann graduated from the Gymnasium in March 1855 but his health at this stage was not good so, in order to recover from tuberculosis, he was advised to spend the summer in Switzerland which he did before beginning his university education.
- Bachmann entered the University of Berlin (the Friedrich-Wilhelms-Universität) in the autumn of 1855 and studied mathematics there in the Faculty of Philosophy.
- In Göttingen, Bachmann became close friends with Dedekind who had only a few years earlier been awarded his doctorate under Gauss's supervision.
- At the University of Göttingen Bachmann attended courses by Wilhelm Weber, Friedrich Woehler, Hermann Lotze, Bernhard Riemann and Richard Dedekind.
- Bachmann returned to Berlin where he received a doctorate in 1862 for a thesis on group theory under Eduard Kummer's supervision.
- His dissertation "De substitutionum theoria meditationes quaedam" Ⓣ(On substitution theory of any the kind) was examined by E Fischer, F Bachmann and J Teichert on 24 March 1862.
- From Berlin, Bachmann went to Breslau to study for his habilitation.
- Freed from administrative tasks, Bachmann was able to carry out his plans for major publications that he had been thinking about for a long time.
- Bachmann followed this by: Volume 2, "Die analytische Zahlentheorie" Ⓣ(Analytic number theory) (1894); Volume 3, "Die Lehre von der Kreisteilung und ihr Beziehungen zur Zahlentheorie" Ⓣ(The theory of cyclotomic polynomials and their relations with the number theory) (1872); Volume 4, "Die Arithmetik der quadratischen Formen" Ⓣ(The arithmetic of quadratic forms) (Part 1, 1898, Part 2, 1923); and Volume 5, "Allgemeine Arithmetik der Zahlkörper" Ⓣ(General arithmetic of number fields) (1905).
- Bachmann has fully maintained his reputation as to clearness, thoroughness, and exhaustiveness.
- Bachmann surveyed the attempts that had been made over nearly 300 years attempting to give a positive or a negative solution to Fermat's Last Theorem in Das Fermatproblem in Seiner Bisherigen Entwicklung (1919).
- The exposition is unusually detailed, and Bachmann spares no pains to make clear his reasons for attacking various arithmetical problems in the way he does.
- Many mathematicians have a love of music, and some of them they shall exercise in this or that instrument; and some of them they shall practice in this or that instrument; but it is rarely granted to them to be creative in music, as Bachmann's friends know that he was.

Born 22 June 1837, Berlin, Germany. Died 31 March 1920, Weimar, Germany.

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Algebra, Origin Germany

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive