Commit 06c8b069 by 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} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End:
 \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} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End:
 \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} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End:
 \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{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 as $\infinitesum{n}1{\nPrimeNumber{n} \power{10}{\uminus{n+ \Sumfromto{k}1n{\floor{\logten{\nPrimeNumber{k}}}}}}}$ \end{definition}