Modular Arithmetics before C.F. Gauss. Maarten Bullynck To cite this version: Maarten Bullynck. Modular Arithmetics before C.F. Gauss.: Systematizations and discussions on remainder problems in 18th-century Germany.. Historia Mathematica, Elsevier, 2009, 36 (1), pp.48- 72. halshs-00663292 HAL Id: halshs-00663292 https://halshs.archives-ouvertes.fr/halshs-00663292 Submitted on 26 Jan 2012 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Modular Arithmetic before C.F. Gauss. Systematisations and discussions on remainder problems in 18th century Germany Maarten Bullynck IZWT, Bergische Universit¨at Wuppertal, Gaußstraße 20, 42119 Wuppertal, Germany Forschungsstipendiat der Alexander-von-Humboldt-Stiftung Email:
[email protected] Abstract Remainder problems have a long tradition and were widely disseminated in books on calculation, algebra and recreational mathematics from the 13th century until the 18th century. Many singular solution methods for particular cases were known, but Bachet de M´eziriac was the first to see how these methods connected with the Euclidean algorithm and with Diophantine analysis (1624). His general solution method contributed to the theory of equations in France, but went largely unno- ticed elsewhere. Later Euler independently rediscovered similar methods, while von Clausberg generalised and systematised methods which used the greatest common divisor procedure.