Geodesic
An elementary proof of a congruence by Skula and Granville
Archivum mathematicum, Tome 48 (2012) no. 2, pp. 113-120

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

Let $p\ge 5$ be a prime, and let $q_p(2):=(2^{p-1}-1)/p$ be the Fermat quotient of $p$ to base $2$. The following curious congruence was conjectured by L. Skula and proved by A. Granville \[ q_p(2)^2\equiv -\sum _{k=1}^{p-1}\frac{2^k}{k^2}\quad(\operatorname{mod} p)\,. \] In this note we establish the above congruence by entirely elementary number theory arguments.
DOI : 10.5817/AM2012-2-113
Classification : 05A10, 11A07, 11B65
Keywords: congruence; Fermat quotient; harmonic numbers 
@article{10_5817_AM2012_2_113,
     author = {Me\v{s}trovi\'c, Romeo},
     title = {An elementary proof of a congruence by {Skula} and {Granville}},
     journal = {Archivum mathematicum},
     pages = {113--120},
     publisher = {mathdoc},
     volume = {48},
     number = {2},
     year = {2012},
     doi = {10.5817/AM2012-2-113},
     mrnumber = {2946211},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5817/AM2012-2-113/}
}
TY  - JOUR
AU  - Meštrović, Romeo
TI  - An elementary proof of a congruence by Skula and Granville
JO  - Archivum mathematicum
PY  - 2012
SP  - 113
EP  - 120
VL  - 48
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.5817/AM2012-2-113/
DO  - 10.5817/AM2012-2-113
LA  - en
ID  - 10_5817_AM2012_2_113
ER  - 
%0 Journal Article
%A Meštrović, Romeo
%T An elementary proof of a congruence by Skula and Granville
%J Archivum mathematicum
%D 2012
%P 113-120
%V 48
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.5817/AM2012-2-113/
%R 10.5817/AM2012-2-113
%G en
%F 10_5817_AM2012_2_113
Meštrović, Romeo. An elementary proof of a congruence by Skula and Granville. Archivum mathematicum, Tome 48 (2012) no. 2, pp. 113-120. doi: 10.5817/AM2012-2-113

Cité par Sources :