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.
%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
}