46.193.64.90: typo correction in Signature verification

2013-05-01T10:17:07Z

typo correction in Signature verification

← Older revision		Revision as of 12:17, 1 May 2013
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	50 year ~~old Wine Maker Brickley from Lac du Bonnet~~, ~~spends time with hobbies for instance jigsaw puzzles, Marvel Puzzle Quest Hack~~ and ~~fitness~~. ~~Has signed up for~~ a ~~world contiki trip. Is very excited specifically about traveling to Las Médulas.~~<br><br>~~Stop~~ by ~~my blog post~~ [http://~~www~~.~~pinterest~~.~~com~~/~~pin~~/~~420805158906729153~~/ ~~Marvel Puzzle Quest Cheats~~]		In mathematics, '''Kummer's congruences''' are some [[Congruence relation\|congruence]]s involving [[Bernoulli numbers]], found by {{harvs\|txt\|authorlink=Ernst Eduard Kummer\|first=Ernst Eduard\|last= Kummer\|year=1851}}.

			{{harvtxt\|Kubota\|Leopoldt\|1964}} used Kummer's congruences to define the [[p-adic zeta function]].

			==Statement==

			The simplest form of Kummer's congruence states that
			:<math> \frac{B_h}{h}\equiv \frac{B_k}{k} \pmod p \text{ whenever } h\equiv k \pmod {p-1}</math>
			where ''p'' is a prime, ''h'' and ''k'' are positive even integers not divisible by ''p''−1 and the numbers ''B''<sub>''h''</sub> are [[Bernoulli numbers]].

			More generally if ''h'' and ''k'' are positive even integers not divisible by ''p'' − 1, then
			:<math> (1-p^{h-1})\frac{B_h}{h}\equiv (1-p^{k-1})\frac{B_k}{k} \pmod {p^{a+1}}</math>
			whenever
			:<math> h\equiv k\pmod {\varphi(p^{a+1})}</math>

			where φ(''p''<sup>''a''+1</sup>) is the [[Euler totient function]], evaluated at ''p''<sup>''a''+1</sup> and ''a'' is a non negative integer. At ''a'' = 0, the expression takes the simpler form, as seen above.
			The two sides of the Kummer congruence are essentially values of the [[p-adic zeta function]], and the Kummer congruences imply that the ''p''-adic zeta function for negative integers is continuous, so can be extended by continuity to all ''p''-adic integers.

			==See also==

			*[[Von Staudt–Clausen theorem]], another congruence involving Bernoulli numbers

			==References==

			*{{Citation \| last1=Koblitz \| first1=Neal \| author1-link=Neal Koblitz \| title=''p''-adic Numbers, ''p''-adic Analysis, and Zeta-Functions \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Graduate Texts in Mathematics, vol. 58 \| isbn=978-0-387-96017-3 \| mr=754003 \| year=1984}}
			*{{Citation \| last1=Kubota \| first1=Tomio \| last2=Leopoldt \| first2=Heinrich-Wolfgang \| title=Eine p-adische Theorie der Zetawerte. I. Einführung der p-adischen Dirichletschen L-Funktionen \| url= http://gdz.sub.uni-goettingen.de/dms/resolveppn/?GDZPPN002180626 \| mr=0163900 \| year=1964 \| journal=[[Journal für die reine und angewandte Mathematik]] \| issn=0075-4102 \| volume=214/215 \| pages=328–339
			\|author1-link=Tomio Kubota\|author2-link=Heinrich-Wolfgang Leopoldt}}
			*{{Citation \| last1=Kummer \| first1=Ernst Eduard \| title=Über eine allgemeine Eigenschaft der rationalen Entwicklungscoëfficienten einer bestimmten Gattung analytischer Functionen \| url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002147319 \| id={{ERAM\|041.1136cj}} \| year=1851 \| journal=Journal für Reine und Angewandte Mathematik \| issn=0075-4102 \| volume= 41 \| pages=368–372}}

			[[Category:Number theory]]

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2011-10-26T13:51:21Z

Bypassing 1 Google redirection URL. Errors? User:AnomieBOT/shutoff/ReplaceExternalLinks4

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50 year old Wine Maker Brickley from Lac du Bonnet, spends time with hobbies for instance jigsaw puzzles, Marvel Puzzle Quest Hack and fitness. Has signed up for a world contiki trip. Is very excited specifically about traveling to Las Médulas.<br><br>Stop by my blog post [http://www.pinterest.com/pin/420805158906729153/ Marvel Puzzle Quest Cheats]

Homomorphic signatures for network coding - Revision history

46.193.64.90: typo correction in Signature verification

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