| Title:
|
An elementary proof of a congruence by Skula and Granville (English) |
| Author:
|
Meštrović, Romeo |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
48 |
| Issue:
|
2 |
| Year:
|
2012 |
| Pages:
|
113-120 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
congruence |
| Keyword:
|
Fermat quotient |
| Keyword:
|
harmonic numbers |
| MSC:
|
05A10 |
| MSC:
|
11A07 |
| MSC:
|
11B65 |
| idMR:
|
MR2946211 |
| DOI:
|
10.5817/AM2012-2-113 |
| . |
| Date available:
|
2012-06-08T08:33:03Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142825 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
|
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| Reference:
|
[6] Granville, A.: Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers.Organic Mathematics–Burnaby, BC 1995, CMS Conf. Proc., vol. 20, Amer. Math. Soc., Providence, RI, 1997, pp. 253–276. Zbl 0903.11005, MR 1483922 |
| Reference:
|
[7] Granville, A.: The square of the Fermat quotient.Integers 4 (2004), # A22. Zbl 1083.11005, MR 2116007 |
| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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[13] Sun, Z. W.: Arithmetic theory of harmonic numbers.Proc. Amer. Math. Soc. 140 (2012), 415–428. MR 2846311, 10.1090/S0002-9939-2011-10925-0 |
| Reference:
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| . |
