%0 Generic %1 citeulike:3036268 %A Euler, Leonhard %D 2007 %K Vor1800 available-in-tex-format mathematics number-theory pre1800 %T Theorems on residues obtained by the division of powers %U http://arxiv.org/abs/math/0608467 %X This is an English translation of Euler's ``Theoremata circa residua ex divisione potestatum relicta'', Novi Commentarii academiae scientiarum Petropolitanae 7 (1761), 49-82. E262 in the Enestrom index. Euler gives many elementary results on power residues modulo a prime number p. He shows that the order of a subgroup generated by an element a in F\_p^* must divide the order p-1 of F\_p^* (i.e. a special case of Lagrange's theorem for cyclic groups). Euler also gives a proof of Fermat's little theorem, that a^p-1 = 1 mod p for a relatively prime to p (i.e. not 0 mod p). He remarks that this proof is more natural, as it uses multiplicative properties of F\_p^* instead of the binomial expansion. Thanks to Jean-Marie Bois for pointing out some typos.

@misc{citeulike:3036268, abstract = {This is an English translation of Euler's ``Theoremata circa residua ex divisione potestatum relicta'', Novi Commentarii academiae scientiarum Petropolitanae 7 (1761), 49-82. E262 in the Enestrom index. Euler gives many elementary results on power residues modulo a prime number p. He shows that the order of a subgroup generated by an element a in F\_p^* must divide the order p-1 of F\_p^* (i.e. a special case of Lagrange's theorem for cyclic groups). Euler also gives a proof of Fermat's little theorem, that a^{p-1} = 1 mod p for a relatively prime to p (i.e. not 0 mod p). He remarks that this proof is more natural, as it uses multiplicative properties of F\_p^* instead of the binomial expansion. Thanks to Jean-Marie Bois for pointing out some typos.}, added-at = {2009-08-02T17:14:35.000+0200}, archiveprefix = {arXiv}, author = {Euler, Leonhard}, biburl = {https://www.bibsonomy.org/bibtex/220030d83425840659304bfff9d724273/rwst}, citeulike-article-id = {3036268}, citeulike-linkout-0 = {http://arxiv.org/abs/math/0608467}, citeulike-linkout-1 = {http://arxiv.org/pdf/math/0608467}, description = {my bookmarks from citeulike}, eprint = {math/0608467}, interhash = {93c62efc3a7f1aa5d98c9135535d8804}, intrahash = {20030d83425840659304bfff9d724273}, keywords = {Vor1800 available-in-tex-format mathematics number-theory pre1800}, month = Aug, posted-at = {2008-07-23 08:43:24}, priority = {2}, timestamp = {2009-08-06T10:18:29.000+0200}, title = {Theorems on residues obtained by the division of powers}, url = {http://arxiv.org/abs/math/0608467}, year = 2007 }