Commit 06c8b069 authored by Michael Kohlhase's avatar Michael Kohlhase

moved modules to new repositories

parent 909d398c
id: smglom/smglom
source-base: http://mathhub.info/smglom/smglom
narration-base: http://mathhub.info/smglom/smglom
dependencies: smglom/linear-algebra,smglom/trigonometry,smglom/calculus,smglom/analysis,smglom/numberfields,smglom/sets,smglom/algebra,smglom/primes,smglom/mv,smglom/numbers
dependencies: smglom/linear-algebra,smglom/numthyfun,smglom/calculus,smglom/numberfields,smglom/sets,smglom/algebra,smglom/primes,smglom/mv,smglom/numbers
responsible: juergen.schefter@Zentralblatt-MATH.org
title: Staging Ground
teaser: Various mathematical concepts to be sorted into SMGloM repositories
......
\begin{modnl}[creators=jusche]{abcconjecture}{de}
\begin{definition}
\defi[ABC-conjecture]{abc-Vermutung}: F"ur jedes reelle $\varepsilon > 0$ existiert eine
Konstante $\livar{K}\varepsilon$, sodass f"ur alle Tripel
\mtrefi[coprime?coprime]{teilerfremder}
\mtrefi[naturalnumbers?natural-number]{nat"urlicher Zahlen} $\tup{a,b,c}$ mit $a+b=c$
die folgende Ungleichung gilt:
\[c < \atimes{\livar{K}{\varepsilon},\power{\radint{\nattimes{a,b,c}}}{1+\varepsilon}}\]
\end{definition}
\end{modnl}
\begin{modnl}[creators=jusche]{abcconjecture}{en}
\begin{definition}
\defii[ABC-conjecture]{ABC}{Conjecture}: For every $\varepsilon > 0$, there exist only
finitely many triples $\tup{a,b,c}$ of \trefi[coprime]{coprime}
\trefii[naturalnumbers]{natural}{number}s, with $a + b = c$, such that
\[c>\radint{\power{\nattimes{a,b,c}}{1+\varepsilon}}\]
\end{definition}
\begin{definition}
\defii[ABC-conjecture]{ABC}{Conjecture}: For every $\varepsilon > 0$, there exists a
constant $\livar{K}\varepsilon$ for all triples $\tup{a, b, c}$ of coprime positive
integers, with $a + b = c$, such that
\[c < \atimes{\livar{K}{\varepsilon},\power{\radint{abc}}{1+\varepsilon}}\]
\end{definition}
\begin{definition}
\defii[ABC-conjecture]{ABC}{Conjecture}: For every $\varepsilon > 0$, there exist only
finitely many triples $\tup{a,b,c}$ of coprime positive integers with $a + b = c$ such
that
\[\nappa{q}{a,b,c} = \frac{\natlog{c}}{\natlog{\radint{\nattimes{a,b,c}}}} > 1 + \varepsilon\]
\end{definition}\ednote{MK: many definitions for the same thing, needs to be split and
viewed.}
\end{modnl}
\begin{modsig}[creators=jusche]{abcconjecture}
\gimport[smglom/sets]{cartesian-product}
\gimport[smglom/primes]{coprime}
\gimport[smglom/numberfields]{natarith}
\gimport[smglom/calculus]{common-logarithm}
\gimport[smglom/calculus]{naturallogarithm}
\gimport{radicalofaninteger}
\symii{ABC}{conjecture}
\end{modsig}
\begin{modnl}[creators=jusche]{abundance}{de}
\begin{definition}
Die \defi[abundance]{Abundanz} einer positiven
\mtrefii[integernumbers?integer]{ganzen}{Zahl} $n$ ist der Wert
$\aminus{\sumdiv{n},\atimes{2,n}}$.
\end{definition}
\end{modnl}
\begin{modnl}[creators=jusche]{abundance}{en}
\begin{definition}
The \defi{abundance} of a positive \trefi[integernumbers]{integer} $n$ is the value
$\aminus{\sumdiv{n},\atimes{2,n}}$.
\end{definition}
\end{modnl}
\begin{modsig}[creators=jusche]{abundance}
\gimport[smglom/numberfields]{integernumbers}
\gimport{sumofdivisorsfunction}
\symi{abundance}
\end{modsig}
\begin{modnl}[creators=jusche]{abundancy}{de}
\begin{definition}
Die \defi[abundancy]{Abundancy} einer \mtrefi[integernumbers?integer]{ganzen Zahl} $n$
ist das Verh\"altnis $\sumdiv{n}/n$. Dabei ist $\sumdiv{n}$ die
\mtrefi[sumofdivisorsfunction?sumdiv-function]{Teilersummenfunktion}.
\end{definition}
\end{modnl}
\begin{modnl}[creators=jusche]{abundancy}{en}
\begin{definition}
The \defi{abundancy} of a \atrefi[integernumbers]{number}{integer} $n$ is defined as the
ratio $\sumdiv{n}/n$, where $\sumdiv{n}$ is the
\atrefi[sumofdivisorsfunction]{sum-of-divisors function}{sumdiv-function}.
\end{definition}
\end{modnl}
\begin{modsig}[creators=jusche]{abundancy}
\gimport[smglom/numberfields]{integernumbers}
\gimport{sumofdivisorsfunction}
\symi{abundancy}
\end{modsig}
\begin{modnl}[creators=jusche]{aliquotsequence}{de}
\begin{definition}
Eine \defi[aliquot-sequence]{Aliquot-Folge} startet mit einer positiven
\mtrefii[integernumbers?integer]{ganzen}{Zahl} $k$. Jedes weitere Glied ist die Summe
der \mtrefii[divisor?divisor]{echten}{Teiler} des vorhergehenden Gliedes.
\end{definition}
\end{modnl}
\begin{modnl}[creators=jusche]{aliquotsequence}{en}
\begin{definition}
An \defii{aliquot}{sequence} starting with a positive \trefi[integernumbers]{integer}
$k$ is a recursive \trefi[sequences]{sequence} in which each term is the sum of the
\trefii[divisor]{proper}{divisor}s of the previous term.
\end{definition}
\end{modnl}
\begin{modsig}[creators=jusche]{aliquotsequence}
\gimport[smglom/numberfields]{integernumbers}
\gimport[smglom/numberfields]{divisor}
\gimport[smglom/calculus]{sequences}
\symii{aliquot}{sequence}
\end{modsig}
\begin{modnl}[creators=jusche]{aliquotsum}{de}
\begin{definition}
Die Summe aller echten \mtrefi[divisor?divisor]{Teiler} von $n$ bezeichnet man als
\defi[aliquot-sum]{Teilersumme} und schreibt $\alisum{n}$.
\end{definition}
\end{modnl}
\begin{modnl}[creators=jusche]{aliquotsum}{en}
\begin{definition}
The \defii{aliquot}{sum} $\alisum{n}$ is defined as the sum of the
\trefii[divisor]{proper}{divisor}s of $n$.
\end{definition}
\end{modnl}
\begin{modsig}[creators=jusche]{aliquotsum}
\gimport[smglom/numberfields]{divisor}
\symdef[name=aliquot-sum]{alisumOp}{\text{s}}
\symdef[name=aliquot-sum]{alisum}[1]{\prefix\alisumOp{#1}}
\symtest{alisum}{\alisum{8}=1+2+4}
\end{modsig}
\begin{modnl}[creators=jusche]{barrier}{de}
\begin{definition}
Eine \mtrefii[realnumbers?real-number]{reelle}{Zahl} $n$ wird \defi[barrier]{Barriere}
einer
\mtrefii[numbertheoreticfunction?arithmeticfunction]{zahlentheoretischen}{Funktion}
$f(m)$ genant, wenn $m+f(m)\le n$ f"ur alle $m>n$.
\end{definition}
\end{modnl}
\begin{modnl}[creators=jusche]{barrier}{en}
\begin{definition}
A \atrefii[realnumbers]{number}{real}{number} $n$ is called a \defi{barrier} of a
\atrefi[numbertheoreticfunction]{number-theoretic function}{arithmeticfunction} $f(m)$
if, for all $m>n$, $m+f(m)\le n$.
\end{definition}
\end{modnl}
\begin{modsig}[creators=jusche]{barrier}
\gimport[smglom/numberfields]{realnumbers}
\gimport{numbertheoreticfunction}
\symi{barrier}
\end{modsig}
\begin{modnl}[creators=jusche]{berahaconstants}{de}
\begin{definition}
Die $n$-te \defi[Beraha-constant]{Beraha-Konstante} ist definiert als
\[\fundefeq{n}{\berahaconstant{n}}{2+2\cosine{\frac{2\pinumber}{n}}}\]
\end{definition}
\end{modnl}
\begin{modnl}[creators=jusche]{berahaconstants}{en}
\begin{definition}
The $n$th \defii{Beraha}{constant} is defind by
\[\fundefeq{n}{\berahaconstant{n}}{2+2 \cosine{\frac{2\pinumber}{n}}}\]
\end{definition}
\end{modnl}
\begin{modsig}[creators=jusche]{berahaconstants}
\gimport[smglom/sets]{fundefeq}
\gimport[smglom/numberfields]{pinumber}
\gimport[smglom/trigonometry]{cosine-series}
\symdef[name=Beraha-constant]{berahaconstantOp}{\mathop{B}}
\symdef[name=Beraha-constant]{berahaconstant}[1]{\prefix\berahaconstantOp{#1}}
\symtest{berahaconstant}{\berahaconstant{1}=4}
\end{modsig}
\begin{modnl}[creators=miko]{binomialcoefficient}{de}
\begin{definition}
Der \adefii{Binomialkoeffizient}{binomial}{coefficient} $\binomcoeff{n}{k}$ ist
definiert als die Anzahl der $k$-elementigen Teilmengen einer $n$-elementigen Menge.
\end{definition}
\end{modnl}
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\begin{modnl}[creators=miko]{binomialcoefficient}{en}
\begin{definition}
The \defii{binomial}{coefficient} $\binomcoeff{n}{k}$ (written as
$\binomcoeff[fr]{n}{k}$ in French) is defined to be the number of $k$-element subsets of
an $n$-element set.
\end{definition}
\end{modnl}
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\begin{modsig}[creators=miko]{binomialcoefficient}
\gimport[smglom/sets]{finite-cardinality}
\symdef[name=binomial-coefficient]{binomcoeff}[2]{{#1\choose #2}}
\symvariant{binomcoeff}[2]{fr}{\mathop{\mathcal{C}}^{#1}_{#2}}
\end{modsig}
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\begin{modnl}[creators=jusche]{brunconstant}{de}
\begin{definition}
Die Summe der Kehrwerte aller \mtrefi[primetwin?prime-twin]{Primzahlzwillinge}
konvergiert. Den Grenzwert bezeichnet man als
\defii[Bruns-constant]{Brunsche}{Konstante} $\brunconstant$.
\[\defeq\brunconstant{\SumProp{p}{\text{$p$, $p+2$ prim}}{\frac1{p}+\frac1{p+2}}}\]
\[\iswitherror\brunconstant{\decrealnum[comma]{1}{902160583209}}{\decrealnum[comma]{0}{000000000781}}\]
\end{definition}
\end{modnl}
\begin{modnl}[creators=jusche]{brunconstant}{en}
\begin{definition}
The sum of the reciprocals of the \atrefii[primetwin]{twin primes}{prime}{twin}
converges to a finite value, the \defii[Bruns-constant]{Brun's}{constant}
$\brunconstant$.
\[\defeq\brunconstant{\SumProp{p}{\text{$p$, $p+2$ prime}}{\aplus{\frac1{p},\frac1{p+2}}}}\]
\[\iswitherror\brunconstant{\decrealnum{1}{902160583209}}{\decrealnum{0}{000000000781}}\]
\end{definition}
\end{modnl}
\begin{modsig}[creators=jusche]{brunconstant}
\gimport[smglom/primes]{primetwin}
\gimport[smglom/numberfields]{witherror}
\gimport[smglom/numberfields]{sum}
\gimport[smglom/numberfields]{realpns}
\symdef[name=Bruns-constant]{brunconstant}{\mathop{B_2}}
\symtest{brunconstant}{\startswith\brunconstant{\decrealnum{\decrealnum{1}{90216}}}}
\end{modsig}
\begin{modnl}[creators=jusche]{brunconstantcousin}{de}
\begin{definition}
Die Summe der Kehrwerte aller \mtrefi[cousinprime?Cousin-prime]{Cousin-Primzahlen}
konvergiert. Den Grenzwert bezeichnet man als
\defiii[Bruns-constant-cousin]{Brunsche}{Konstante}{f"ur Cousin-Primzahlen}
$\brunconstantcousin$.
\[\defeq\brunconstantcousin{\aplus{\frac{1}{7} + \frac{1}{11},\frac{1}{13} + \frac{1}{17},
\frac{1}{19} + \frac{1}{23},\cdots}}
\]
$\startswith\brunconstantcousin{\decrealnum[comma]{1}{1970449}}$
\end{definition}
\end{modnl}
\begin{modnl}[creators=jusche]{brunconstantcousin}{en}
\begin{definition}
The sum of the reciprocals of the \trefii[cousinprime]{Cousin}{prime}s converges to a
finite value, the \defiii[Bruns-constant-cousin]{Brun's}{constant}{for cousin primes}
$\brunconstantcousin$.
\[\defeq\brunconstantcousin{\aplus{\frac{1}{7} + \frac{1}{11},\frac{1}{13} + \frac{1}{17},
\frac{1}{19} + \frac{1}{23},\cdots}}\]
\[\startswith\brunconstantcousin{\decrealnum{1}{1970449}}\]
\end{definition}
\end{modnl}
\begin{modsig}[creators=jusche]{brunconstantcousin}
\gimport[smglom/primes]{cousinprime}
\gimport[smglom/primes]{primetwin}
\gimport[smglom/numberfields]{realpns}
\symdef[name=Bruns-constant-cousin]{brunconstantcousin}{B_4}
\symtest{brunconstantcousin}{\brunconstantcousin \approx \decrealnum{1}{197}}
\end{modsig}
\begin{modnl}[creators=jusche]{brunconstantquad}{de}
\begin{definition}
Die Summe der Kehrwerte aller
\mtrefi[primequadruplet?prime-quadruplet]{Primzahlvierlinge} konvergiert. Den Grenzwert
bezeichnet man als \defiii[Bruns-constant-quad]{Brunsche}{Konstante}{f"ur
Primzahlvierlinge} $\brunconstantquad$.
\[\defeq\brunconstantquad{\aplus{\frac{1}{5} + \frac{1}{7} + \frac{1}{11} + \frac{1}{13},
\frac{1}{11} + \frac{1}{13} + \frac{1}{17} + \frac{1}{19},
\frac{1}{101} + \frac{1}{103} + \frac{1}{107} + \frac{1}{109},\cdots}}\]
\[\iswitherror{\brunconstantquad}{\decrealnum[comma]{0}{8705883800}}{\decrealnum[comma]{0}{00000
00005}}\]
\end{definition}
\end{modnl}
\begin{modnl}[creators=jusche]{brunconstantquad}{en}
\begin{definition}
The sum of the reciprocals of the \trefii[primequadruplet]{prime}{quadruplet}s converges
to a finite value, the \defiii[Bruns-constant-quad]{Brun's}{constant}{for prime
quadruplets} $\brunconstantquad$.
\[\defeq\brunconstantquad{\aplus{\frac{1}{5} + \frac{1}{7} + \frac{1}{11} + \frac{1}{13},
\frac{1}{11} + \frac{1}{13} + \frac{1}{17} + \frac{1}{19},
\frac{1}{101} + \frac{1}{103} + \frac{1}{107} + \frac{1}{109},\cdots}}\]
\[\iswitherror{\brunconstantquad}{\decrealnum{0}{8705883800}}{\decrealnum{0}{00000 00005}}\]
\end{definition}
\end{modnl}
\begin{modsig}[creators=jusche]{brunconstantquad}
\gimport[smglom/numberfields]{realnumbers}
\gimport[smglom/numberfields]{witherror}
\gimport[smglom/primes]{primequadruplet}
\gimport[smglom/numberfields]{realpns}
\symdef[name=Bruns-constant-quad]{brunconstantquad}{B_4}
\symtest{brunconstantquad}{\startswith\brunconstantquad{\decrealnum{0}{87058}}}
\end{modsig}
\begin{modnl}[creators=jusche]{cahenconstant}{de}
\begin{definition}
Sei $\sylvester{n}$ die \mtrefi[sylvestersequence?Sylvester-sequence]{Sylvester-Folge}.
Die \defii[cahenconstant]{Cahen-Konstante} ist eine unendliche
\mtrefi[series?series]{Reihe} von \mtrefi[unitfraction?unit-fraction]{Stammbr"uchen} mit
alternierenden Vorzeichen
\[\defeq\cahenconst{\infinitesum{i}0{\frac{\power{(-1)}i}{\sylvester{i}-1}}}\]
\end{definition}
\end{modnl}
\begin{modnl}[creators=jusche]{cahenconstant}{en}
\begin{definition}
Let $\sylvester{n}$ be the \trefii[sylvestersequence]{Sylvester}{sequence}.
\defii[cahenconstant]{Cahen's}{constant} is defined as an infinite
\trefi[series]{series} of \trefii[unitfraction]{unit}{fraction}s with alternating signs
\[\defeq\cahenconst{\infinitesum{i}0{\frac{\power{(-1)}i}{\sylvester{i}-1}}}\]
\end{definition}
\end{modnl}
\begin{modsig}[creators=jusche]{cahenconstant}
\gimport[smglom/calculus]{infinitesum}
\gimport{unitfraction}
\gimport[smglom/numbers]{sylvestersequence}
\gimport[smglom/numberfields]{realpns}
\symdef[name=cahenconstant]{cahenconst}{\mathop{C}}
\symtest{cahenconst}{\startswith\cahenconst{\decrealnum{0}{64341}}}
\end{modsig}
\begin{modnl}[creators=jusche]{cahenconstantegyptian}{de}
\begin{definition}
Sei $\sylvester{n}$ die \mtrefi[sylvestersequence?Sylvester-sequence]{Sylvester-Folge}.
Die \defi[cahenconstantegyptian]{Cahen-Konstante} $\cahenconstegyptian$ ist ein
\mtrefii[egyptianfraction?Egyptian-fraction]{"agyptischer}{Bruch}, gebildet aus einer
unendlichen \mtrefi[series?series]{Reihe} von
\mtrefi[unitfraction?unit-fraction]{Stammbr"uchen}, deren Nenner die geradzahligen
Elemente der Sylvester-Folge sind:
\[\defeq\cahenconstegyptian{\infinitesum{i}0{\frac{1}{\sylvester{{2i}}}}}\]
Wir haben also
\[\cahenconstegyptian=\frac12+\frac17+\frac1{1807}+\frac1{10650056950807}+\cdots\]
und damit $\startswith\cahenconstegyptian{\decrealnum[comma]{0}{64341054629}}$
\end{definition}
\end{modnl}
\begin{modnl}[creators=jusche]{cahenconstantegyptian}{en}
\begin{definition}
Let $\sylvester{n}$ be the \trefii[sylvestersequence]{Sylvester}{sequence}.
\defii[cahenconstantegyptian]{Cahen's}{constant} $\cahenconstegyptian$ is an
\trefii[egyptianfraction]{Egyptian}{fraction} formed by an infinite
\trefi[series]{series} of \trefii[unitfraction]{unit}{fraction}s, which denominators are
the even elements of Sylvester's sequence:
\[\defeq\cahenconstegyptian{\infinitesum{i}0{\frac{1}{\sylvester{{2i}}}}}\]
We have
\[\cahenconstegyptian=\frac12+\frac17+\frac1{1807}+\frac1{10650056950807}+\cdots\]
and thus $\startswith\cahenconstegyptian{\decrealnum{0}{64341054629}}$
\end{definition}
\end{modnl}
\begin{modsig}[creators=jusche]{cahenconstantegyptian}
\gimport[smglom/calculus]{infinitesum}
\gimport{unitfraction}
\gimport[smglom/numbers]{sylvestersequence}
\gimport{egyptianfraction}
\gimport[smglom/mv]{defeq}
\gimport[smglom/numberfields]{realpns}
\symdef{cahenconstegyptian}{C}
\symtest{cahenconstegyptian}{\startswith\cahenconstegyptian{\decrealnum{0}{64341}}}
\symi{cahenconstantegyptian}
\end{modsig}
\begin{modnl}[creators=jusche]{cassiniidentity}{de}
\begin{definition}
Die \defi[Cassini-identity]{Cassini-Identit"at} ist eine Identit"at f"ur
\mtrefii[integernumbers?integers]{ganze}{Zahlen} $n > 0$ und die
\mtrefi[fibonaccinumbers?Fibonacci-numbers]{Fibonacci-Zahlen}.
\[\fibnum{n-1}\fibnum{n+1} - \power{\fibnum{n}}2 = \power{(-1)}n\]
\end{definition}
\end{modnl}
\begin{modnl}[creators=jusche]{cassiniidentity}{en}
\begin{definition}
The \defii{Cassini}{identity} is an identity for \trefi[integernumbers]{integers} $n >
0$ and the \trefii[fibonaccinumbers]{Fibonacci}{numbers}.
\[\fibnum{n-1}\fibnum{n+1} - \power{\fibnum{n}}2 = \power{(-1)}n\]
\end{definition}
\end{modnl}
\begin{modsig}[creators=jusche]{cassiniidentity}
\gimport[smglom/numbers]{fibonaccinumbers}
\gimport[smglom/numberfields]{integernumbers}
\gimport[smglom/numberfields]{arithmetics}
\symii{Cassini}{identity}
\end{modsig}
\begin{modnl}[creators=jusche]{catalanidentity}{de}
\begin{definition}
Die \defi[Catalan-identity]{Catalan-Identit"at} ist eine Identit"at f"ur
\mtrefii[integernumbers?integers]{ganze}{Zahlen} $n > r > 0$, und die
\mtrefi[fibonaccinumbers?Fibonacci-numbers]{Fibonacci-Zahlen}.
\[\power{\fibnum{n}}2-\fibnum{n-r}\fibnum{n+r} = \power{(-1)}{n-r}\power{\fibnum{r}}2\]
\end{definition}
\end{modnl}
\begin{modnl}[creators=jusche]{catalanidentity}{en}
\begin{definition}
The \defii{Catalan}{identity} is an identity for \trefi[integernumbers]{integers} $n > r
> 0$, and the \trefii[fibonaccinumbers]{Fibonacci}{numbers}.
\[\power{\fibnum{n}}2-\fibnum{n-r}\fibnum{n+r} = \power{(-1)}{n-r}\power{\fibnum{r}}2\]
\end{definition}
\end{modnl}
\begin{modsig}[creators=jusche]{catalanidentity}
\gimport[smglom/numbers]{fibonaccinumbers}
\gimport[smglom/numberfields]{integernumbers}
\gimport[smglom/numberfields]{arithmetics}
\symii{Catalan}{identity}
\end{modsig}
\begin{modnl}[creators=jusche]{champerowneconstant}{de}
\begin{definition}
Die \defi[Champerowne-constant]{Champerowne-Zahl} $\champerowneconst$ ist eine
\mtrefi[normalnumber?normal]{normale} Zahl der Basis $10$.
Sie wird durch das Aneinanderreihen der
\mtrefii[naturalnumbers?natural-number]{nat"urlichen}{Zahlen} gebildet.
\[\defeq\champerowneconst
{\infinitesum{n}1{\Sumfromto{k}{\power{10}{n-1}}{\power{10}n-1}{\frac{k}{\power{10}{n(k-\power{10}{n-1}+1)+
9\Sumfromto{l}1{n-1}{\power{10}{l-1}l}}}}}}\]
Champernowne-Zahlen k"onnen auch in anderen Basen konstruiert werden.
\end{definition}
\begin{omtext}
$\startswith\champerowneconst{\decrealnum[comma]{0}{12345678910111213141516}}$
\end{omtext}
\end{modnl}
\begin{modnl}[creators=jusche]{champerowneconstant}{en}
\begin{definition}
The \defii{Champernowne}{constant} $\champerowneconst$ is a \trefi[normalnumber]{normal}
number for base $10$.
It is defined by concatenating representations of successive \trefi[integernumbers]{integers}:
\[\defeq\champerowneconst
{\infinitesum{n}1{\Sumfromto{k}{\power{10}{n-1}}
{\power{10}n-1}
{\frac{k}{\power{10}{n(k-\power{10}{n-1}+1)+9
\Sumfromto{l}1{n-1}{\power{10}{l-1}l}}}}}}\]
Champernowne constants can also be constructed in other bases.
\end{definition}
\begin{omtext}
$\startswith\champerowneconst{\decrealnum{0}{12345678910111213141516}}$
\end{omtext}
\end{modnl}
\begin{modsig}[creators=jusche]{champerowneconstant}
\gimport[smglom/numberfields]{naturalnumbers}
\gimport[smglom/analysis]{normalnumber}
\gimport[smglom/calculus]{infinitesum}
\gimport[smglom/numberfields]{realpns}
\symdef[name=Champerowne-constant]{champerowneconst}{C_{10}}
\symtest{champerowneconst}{\startswith\champerowneconst{\decrealnum{0}{12345678910111213}}}
\symii{Champernowne}{constant}
\end{modsig}
\begin{modnl}[creators=jusche]{copelanderdoesconstant}{de}
\begin{definition}
Die \defi[copelanderdoesconstant]{Copeland-Erd"os-Zahl} ist eine
\mtrefi[normalnumber?normal]{normale} Zahl zur Basis $10$. Sie wird durch die
Aneinanderreihung der \mtrefi[primenumber?prime-number]{Primzahlen} gebildet.
\[\decrealnum[comma]{0}{235711131719232931374143\dots}\]
Sei $\nPrimeNumber{n}$ die $n$-te Primzahl. Dann erh"alt man f"ur die Copeland-Erd"os-Zahl
\[\infinitesum{n}1{\nPrimeNumber{n} \power{10}{\uminus{n+
\Sumfromto{k}1n{\floor{\logten{\nPrimeNumber{k}}}}}}}\]
\end{definition}
\end{modnl}
\begin{modnl}[creators=jusche]{copelanderdoesconstant}{en}
\begin{definition}
The \defii[copelanderdoesconstant]{Copeland-Erd\"os}{constant} is a
\trefi[normalnumber]{normal} number for base $10$ defind by the representations of the
\trefii[primenumber]{prime}{number}s in order.
\[\decrealnum{0}{235711131719232931374143} \dots\]
If $\nPrimeNumber{n}$ is the $n$th prime number. Then we obtain the Copeland-Erd\"os-Zahl