User talk:Musictheory2math

Whole my notes is visible in the revision as of 18:42, 13 April 2018 Alireza Badali 21:52, 13 April 2018 (CEST)

algebraic topology $\color{fuchsia}{\times}$ number theory

Lemma: For each subinterval $(a,b)$ of $[0.1,1),\,\exists m\in \Bbb N$ that $\forall k\in \Bbb N$ with $k\ge m$ then $\exists t\in (a,b)$ that $t\cdot 10^k\in \Bbb P$.

Proof given by @Adayah from stackexchange.com: Without loss of generality (by passing to a smaller subinterval) we can assume that $(a, b) = \left( \frac{s}{10^r}, \frac{t}{10^r} \right)$, where $s, t, r$ are positive integers and $s < t$. Let $\alpha = \frac{t}{s}$.

The statement is now equivalent to saying that there is $m \in \mathbb{N}$ such that for every $k \geqslant m$ there is a prime $p$ with $10^{k-r} \cdot s < p < 10^{k-r} \cdot t$.

We will prove a stronger statement: there is $m \in \mathbb{N}$ such that for every $n \geqslant m$ there is a prime $p$ such that $n < p < \alpha \cdot n$. By taking a little smaller $\alpha$ we can relax the restriction to $n < p \leqslant \alpha \cdot n$.

Now comes the prime number theorem: $$\lim_{n \to \infty} \frac{\pi(n)}{\frac{n}{\log n}} = 1$$

where $\pi(n) = \# \{ p \leqslant n : p$ is prime$\}.$ By the above we have $$\frac{\pi(\alpha n)}{\pi(n)} \sim \frac{\frac{\alpha n}{\log(\alpha n)}}{\frac{n}{\log(n)}} = \alpha \cdot \frac{\log n}{\log(\alpha n)} \xrightarrow{n \to \infty} \alpha$$

hence $\displaystyle \lim_{n \to \infty} \frac{\pi(\alpha n)}{\pi(n)} = \alpha$. So there is $m \in \mathbb{N}$ such that $\pi(\alpha n) > \pi(n)$ whenever $n \geqslant m$, which means there is a prime $p$ such that $n < p \leqslant \alpha \cdot n$, and that is what we wanted.♦

Now we can define function $f:\{(c,d)\mid (c,d)\subseteq [0.01,0.1)\}\to\Bbb N$ that $f((c,d))$ is the least $n\in\Bbb N$ that $\exists t\in(c,d),\,\exists k\in\Bbb N$ that $p_n=t\cdot 10^{k+1}$ that $p_n$ is $n$_th prime and $\forall m\ge f((c,d))\,\,\exists u\in (c,d)$ that $u\cdot 10^{m+1}\in\Bbb P$

and $g:(0,0.09)\cap (\bigcup _{k\in\Bbb N} r_k(\Bbb N))\to\Bbb N,$ is a function by $\forall\epsilon\in (0,0.09)\cap (\bigcup _{k\in\Bbb N} r_k(\Bbb N))$ $g(\epsilon)=max(\{f((c,d))\mid d-c=\epsilon,$ $(c,d)\subseteq [0.01,0.1)\})$.

Guess $1$: $g$ isn't an injective function.

Question $1$: Assuming above guess let $[a,a]:=\{a\}$ and $\forall n\in\Bbb N,\, h_n$ is the least subinterval of $[0.01,0.1)$ like $[a,b]$ in terms of size of $b-a$ such that $\{\epsilon\in (0,0.09)\cap (\bigcup _{k\in\Bbb N} r_k(\Bbb N))\mid g(\epsilon)=n\}\subsetneq h_n$ and obviously $g(a)=n=g(b)$ now the question is $\forall n,m\in\Bbb N$ that $m\neq n$ is $h_n\cap h_m=\emptyset$?

Guidance given by @reuns from stackexchange.com:

• For $n \in \mathbb{N}$ then $r(n) = 10^{-\lceil \log_{10}(n) \rceil} n$, ie. $r(19) = 0.19$. We look at the image by $r$ of the primes $\mathbb{P}$.
• Let $F((c,d)) = \min \{ p \in \mathbb{P}, r(p) \in (c,d)\}$ and $f((c,d)) = \pi(F(c,d))= \min \{ n, r(p_n) \in (c,d)\}$ ($\pi$ is the prime counting function)
• If you set $g(\epsilon) = \max_a \{ f((a,a+\epsilon))\}$ then try seing how $g(\epsilon)$ is constant on some intervals defined in term of the prime gap $g(p) = -p+\min \{ q \in \mathbb{P}, q > p\}$ and things like $ \max \{ g(p), p > 10^i, p+g(p) < 10^{i+1}\}$

Another guidance: The affirmative answer is given by Liouville's theorem on approximation of algebraic numbers.

Suppose $r:\Bbb N\to (0,1)$ is a function given by $r(n)$ is obtained with putting a point at the beginning of $n$ instance $r(34880)=0.34880$ and similarly consider $\forall k\in\Bbb N\cup\{0\},$ $r_k: \Bbb N \to (0,1)$ given by $\forall n\in\Bbb N$, $r_k(n)=10^{-k}\cdot r(n)$ and let $S=r(\Bbb P)$ and $S_1=\bigcup _{k\in\Bbb N\cup\{0\}}r_k(\Bbb P)$.

Theorem $1$: $S$ is dense in the $[0.1,1]$. (proof using above lemma)

Corollary: For each natural number like $a=a_1a_2a_3...a_k$ that $a_j$ is $j$_th digit for $j=1,2,3,...,k$, there is a natural number like $b=b_1b_2b_3...b_r$ such that the number $c=a_1a_2a_3...a_kb_1b_2b_3...b_r$ is a prime number.

Theorem $2$: $S_1$ is dense in the interval $[0,1]$ and $S_1\times S_1$ is dense in the $[0,1]\times [0,1]$.

An algorithm that makes new cyclic groups on $\Bbb N$:

Let $(\Bbb N,\star)$ be that group and we have $\Bbb N=\langle 2\rangle$ & $e=1$ and at first write integers as a sequence with starting from $0$ for instance: $$0,1,2,-1,-2,3,4,-3,-4,5,6,-5,-6,7,8,-7,-8,9,10,-9,-10,11,12,-11,-12,...$$ $$1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,...$$ then regarding to the sequence find a rotation number that for this sequence is $4$ and hence equations should be written with module $4$, then consider $4m-2,4m-1,4m,4m+1$ that the last should be $km+1$ and initial be $km+(k-2)$ otherwise equations won't match with members inverse definitions, and make a table of productions of those $k$ elements but during writing equations pay attention if an equation is right for given numbers it will be right generally for other numbers for example $12\star 15=6$ or $(4\times 3)\star (4\times 4-1)=6$ because $(-5)+8=3$ & $-5\to 12,\,\, 8\to 15,\,\, 3\to 6,$ that implies $(4n)\star (4m-1)=4m-4n+2$ if $4m-1\gt 4n$ of course it's better at first members inverse be defined for example since $(-9)+9=0$ & $0\to 1,\,\, -9\to 20,\,\, 9\to 18$ so $20\star 18=1$, that implies $(4m)\star (4m-2)=1$, and with a little addition and subtraction all equations will be obtained simply that for this example is:

$\begin{cases} m\star _T 1=m\\ (4m)\star _T (4m-2)=1=(4m+1)\star _T (4m-1)\\ (4m)\star _T (4m+2)=3=(4m+1)\star _T (4m+3)\\ (4m-2)\star _T (4n-2)=4m+4n-5\\ (4m-2)\star _T (4n-1)=4m+4n-2\\ (4m-2)\star _T (4n)=\begin{cases} 4m-4n-1 & 4m-2\gt 4n\\ 4n-4m+1 & 4n\gt 4m-2\end{cases}\\ (4m-2)\star _T (4n+1)=\begin{cases} 4m-4n-2 & 4m-2\gt 4n+1\\ 4n-4m+4 & 4n+1\gt 4m-2\end{cases}\\ (4m-1)\star _T (4n-1)=4m+4n-1\\ (4m-1)\star _T (4n)=\begin{cases} 4m-4n+2 & 4m-1\gt 4n\\ 4n-4m & 4n\gt 4m-1\\ 2 & m=n\end{cases}\\ (4m-1)\star _T (4n+1)=\begin{cases} 4m-4n-1 & 4m-1\gt 4n+1\\ 4n-4m+1 & 4n+1\gt 4m-1\end{cases}\\ (4m)\star _T (4n)=4m+4n-3\\ (4m)\star _T (4n+1)=4m+4n\\ (4m+1)\star _T (4n+1)=4m+4n+1\end{cases}$

Assume $\forall m,n\in\Bbb N$: $\begin{cases} n\star 1=n\\ (2n)\star (2n+1)=1\\ (2n)\star (2m)=2n+2m\\ (2n+1)\star (2m+1)=2n+2m+1\\ (2n)\star (2m+1)=\begin{cases} 2m-2n+1 & 2m+1\gt 2n\\ 2n-2m & 2n\gt 2m+1\end{cases}\end{cases}$

and $p_n\star _1p_m=p_{n\star m}$ that $p_n$ is $n$_th prime, obviously $(\Bbb N,\star)$ & $(\Bbb P,\star _1)$ are groups and $\langle 2\rangle =(\Bbb N,\star)\cong (\Bbb Z,+)\cong (\Bbb P,\star _1)=\langle 3\rangle$.

Suppose $\forall k\in\Bbb N,\, w_k:\Bbb N\to (0,1)$ is a function given by $\forall n\in\Bbb N,\, w_k(n)=r_{k-1}(n)$.

Theorem $3$: Let $e=0.2$ & $\forall w_m(p),w_n(q)\in S_1,\, w_m(p)\star _{S_1} w_n(q)=w_{m\star n} (p\star _1 q)$ that $p,q\in\Bbb P,$ $m,n\in\Bbb N$ & $(w_m(p))^{-1}=w_{m^{-1}}(p^{-1})$ that $m\star m^{-1}=1,\, p\star _1 p^{-1}=2$ now $\langle 0.02,0.3\rangle=(S_1,\star _{S_1})\cong\Bbb Z\oplus\Bbb Z$, of course using above algorithm to generate cyclic groups on $\Bbb N$, we can impose another group structure on $\Bbb N$ and consequently on $\Bbb P$ but eventually $S_1$ with an operation analogous above operation $\star _{S_1}$ will be an Abelian group.

Question $2$: Under which topology will $S_1$ be a topological group?

Theorem $4$: Let $e=(0.2,0.2)$ & $\forall (w_{m_1}(p_1),w_{m_2}(p_2)),(w_{n_1}(q_1),w_{n_2}(q_2))\in S_1\times S_1,$ $$(w_{m_1}(p_1),w_{m_2}(p_2))\star _{S_1\times S_1} (w_{n_1}(q_1),w_{n_2}(q_2))=(w_{m_1\star n_1} (p_1\star _1 q_1),w_{m_2\star n_2}(p_2\star _1 q_2))$$ that $m_1,n_1,m_2,n_2\in\Bbb N,\, p_1,p_2,q_1,q_2\in\Bbb P$ & $(w_m(p),w_n(q))^{-1}=(w_{m^{-1}}(p^{-1}),w_{n^{-1}}(q^{-1}))$ that $m\star m^{-1}=1=n\star n^{-1}$, $p\star _1p^{-1}=2=q\star _1q^{-1}$ now $$\langle (0.02,0.2),(0.2,0.02),(0.3,0.2),(0.2,0.3)\rangle=(S_1\times S_1,\star _{S_1\times S_1})\cong\Bbb Z\oplus\Bbb Z\oplus\Bbb Z\oplus\Bbb Z,$$ of course using above algorithm to generate cyclic groups on $\Bbb N$, we can impose another group structure on $\Bbb N$ and consequently on $\Bbb P$ but eventually $S_1\times S_1$ with an operation analogous above operation $\star _{S_1\times S_1}$ will be an Abelian group.

Question $3$: Under which topology will $S_1\times S_1$ be a topological group?

I want make some topologies having prime numbers properties presentable in collection of open sets, in principle when we image a prime $p$ to real numbers as $w_k(p)$ indeed we accompany prime numbers properties into real numbers which regarding to the expression form of prime number theorem for this aim we should use an important mathematical technique as logarithm function into some planned topologies: question $4$: Let $M$ be a topological space and $A,B$ are subsets of $M$ with $A\subset B$ and $A$ is dense in $B,$ since $A$ is dense in $B,$ is there some way in which a topology on $B$ may be induced other than the subspace topology? I am also interested in specialisations, for example if $M$ is Hausdorff or Euclidean.

Perhaps this technique is useful: $\forall n\in\Bbb N,$ and for each subinterval $(a,b)$ of $[0.1,1),$ that $a\neq b,$

$\begin{cases} U_{(a,b)}:=\{n\in\Bbb N\mid a\le r(n)\le b\},\\ \\V_{(a,b)}:=\{p\in\Bbb P\mid a\le r(p)\le b\},\\ \\U_{(a,b),n}:=\{m\in U_{(a,b)}\mid m\le n\},\\ \\V_{(a,b),n}:=\{m\in V_{(a,b)}\mid m\le n\},\\ \\w_{(a,b),n}:=(\#U_{(a,b),n})^{-1}\cdot\#V_{(a,b),n}\cdot\log n,\\ \\w_{(a,b)}:=\lim _{n\to\infty} w_{(a,b),n}\end{cases}$ ::Guess $2$: $\forall (a,b)\subset [0.1,1),\,w_{(a,b)}=0.9^{-1}\cdot (b-a)$. :::[https://math.stackexchange.com/questions/2683513/an-extension-of-prime-number-theorem/2683561#2683561 Answer] given by [https://math.stackexchange.com/users/82961/peter $@$Peter]: Imagine a very large number $N$ and consider the range $[10^N,10^{N+1}]$. The natural logarithms of $10^N$ and $10^{N+1}$ only differ by $\ln(10)\approx 2.3$ Hence the reciprocals of the logarithms of all primes in this range virtually coincicde. Because of the approximation '"`UNIQ-MathJax8-QINU`"' for the number of primes in the range $[a,b]$ the number of primes is approximately the length of the interval divided by $\frac{1}{\ln(10^N)}$, so is approximately equally distributed. Hence your conjecture is true. :::Benfords law seems to contradict this result , but this only applies to sequences producing primes as the Mersenne primes and not if the primes are chosen randomly in the range above. '''Obviously''' $(S_1,\lt _1)$ is a well-ordering set with order relation $\lt _1$ as: $\forall i,n,k\in\Bbb N$ if $p_n$ be $n$-th prime number, relation $\lt _1$ is defined with: $w_i(p_n)\lt _1w_i(p_{n+k})\lt _1w_{i+1}(p_n)$ or '"`UNIQ-MathJax9-QINU`"' '"`UNIQ-MathJax10-QINU`"' and consequently $(S_1\times S_1,\lt _2)$ will be another well-ordering set with dictionary order relation $\lt _2$ as: $\forall (a,b),(c,d)\in S_1\times S_1,\, (a,b)\lt _2 (c,d)$ if $a\lt _1 c,$ or if $a=c$ and $b\lt _1d$, and assume $M=S_1\times S_1$ is a topological space with order topology induced by $(S_1\times S_1,\lt _2)$. :Question $5$: Does there exist any well known topology on $(0,1)\times (0,1)$ such that $S_1\times S_1$ with subspace topology be homeomorph to the topological space $M$, and is $(S_1\times S_1,\star _{S_1\times S_1})$ a topological group under topology of $M$? Let $t_n:\Bbb N\to\Bbb N\setminus\{n\in\Bbb N\mid 10\mid n\}$ is a surjective sequence that $\forall n\in\Bbb N,\,\,t_n\lt t_{n+1}$ now $\{t_n\}_{n\in\Bbb N}$ is a cyclic group with: $\begin{cases} e=1\\ t_n^{-1}=t_{n^{-1}}\quad\text{that}\quad n\star n^{-1}=1\\ t_n\star _tt_m=t_{n\star m}\end{cases}$ that $(\{t_n\}_{n\in\Bbb N},\star _t)=\langle 2\rangle=\langle 3\rangle$ and $E:=\bigcup _{k\in\Bbb N} w_k(\Bbb N\setminus\{n\in\Bbb N\mid 10\mid n\})$ is an Abelian group with $\forall m,n\in\Bbb N\,\,\forall a,b\in\Bbb N\setminus\{n\in\Bbb N\mid 10\mid n\}$: $\begin{cases} e=0.1\\ w_n(a)^{-1}=w_{n^{-1}}(a^{-1})\quad\text{that}\quad n\star n^{-1}=1,\, a\star _tb=1\\ w_n(a)\star _Ew_m(b)=w_{n\star m}(a\star _tb)\end{cases}$ that $\langle 0.01,0.2\rangle=E\cong\Bbb Z\oplus\Bbb Z$ now assume $(S_1\times S_1)\oplus E$ is external direct product of the groups $S_1\times S_1$ and $(E,\star _E)$ with $e=(0.2,0.2,0.1)$ and $\langle (0.02,0.2,0.1),(0.2,0.02,0.1),(0.3,0.2,0.1),(0.2,0.3,0.1),(0.2,0.2,0.01),(0.2,0.2,0.2)\rangle=$ $(S_1\times S_1)\oplus E\cong\Bbb Z\oplus\Bbb Z\oplus\Bbb Z\oplus\Bbb Z\oplus\Bbb Z\oplus\Bbb Z$. and $(E,\lt _3)$ is a well ordering set with order relation $\lt _3$ as: $\forall i,n,k\in\Bbb N$ that $10\nmid n,\, 10\nmid n+k,$ $w_i(n)\lt w_i(n+k)\lt w_{i+1}(n)$ or '"`UNIQ-MathJax11-QINU`"' '"`UNIQ-MathJax12-QINU`"' now $E$ will be a topological space induced by $(E,\lt _3)$ and assume $D=M\times E$ is a topological space with produced topology. :Question $6$: Is $(S_1\times S_1)\oplus E$ a topological group under topology of $D$? '''A Conjecture''': For each even natural number $t$ greater than $4$ and $\forall c,m\in\Bbb N\cup\{0\}$ that $10^c\mid t,\, 10^{1+c}\nmid t$, $A_m=\{(a,b)\mid a,b\in S_1,\, 10^{-1-m}\le a+b\lt 10^{-m}\}$ and if $u$ is the number of digits in $t$ then $\exists (a,b)\in A_c$ such that $t=10^{c+u}\cdot (a+b),\, 10^{c+u}\cdot a,10^{c+u}\cdot b\in\Bbb P\setminus\{2\},\, (a,b,10^{-c-u}\cdot t)\in D$.

This conjecture is an equivalent to Goldbach's conjecture.

Alireza Badali 08:27, 31 March 2018 (CEST)