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Paul attended a Gymnasium at which he appears to have had some difficulty with mathematics. His mathematical talents, however, came to the fore under the excellent mathematics teacher Karl Schellback and he decided to study mathematics at university. His health at this stage was not good and in order to recover from tuberculosis he was advised to spend some time in Switzerlad, which he did before beginning his university education.

Bachmann entered the University of Berlin and studied mathematics there. In 1856 he went from Berlin to Göttingen so that he could continue to study courses by Dirichlet who had just left Berlin to succeed to Gauss 's chair in Göttingen. In Göttingen Bachmann became close friends with Dedekind who had only a few years earlier been awarded his doctorate under Gauss 's supervision. Bachmann returned to Berlin where he received a doctorate in 1862 for a thesis on group theory under Kummer 's supervision.

From Berlin, Bachmann went to Breslau to study for his habilitation . There he worked on number theory and he was awarded his habilitation in 1864 for a thesis on complex units which had been a topic which he had been inspired to work on though the lectures by Dirichlet which he had attended.

He taught at Breslau after the award of the habilitation, becoming an extraordinary professor there. After a few years he was appointed as full professor at Münster. However Bachamnn divorced his wife in 1890 and resigned his chair in Münster. Then, as Ore writes in :

With his second wife he settled in Weimar, where he combined his mathematical writing with composing, playing the piano, and serving as a music critic for several newspapers.

His most important work is a complete survey of number theory giving both the results and an evaluation of the methods of proof. Zahlentheorie. Versuch einer Gesamtdarstellung dieser Wissenschaft in ihren Hauptteilen which he worked on after resigning his chair in Münster, was published in five volumes between 1892 and 1923. Other major works included Niedere Zahlentheorie published in two volumes in 1902 and 1910 while Das Fermat-Problem in seiner bisherigan Entwicklung was published in 1919.

Source:School of Mathematics and Statistics University of St Andrews, Scotland