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Theorem $1$: Let $\mathbb{P}$ be the set prime numbers and $S$ is a set that has been made as below: put a point on the beginning of each member of $\Bbb{P}$ like $0.2$ or $0.19$ then $S=\{0.2,0.3,0.5,0.7,...\}$ is dense in the interval $(0.1,1)$ of real numbers.

Theorem $2$: For each subinterval of $[0.1,1)$ like $(a,b)$ then $\exists m\in \Bbb N$ that $\forall k\in \Bbb N$ with $k\ge m$ then $\exists t\in (a,b)$ such that $t\cdot 10^k\in \Bbb P$. $$\\$$ Suppose $r:\Bbb N \to (0,1)$ is a function given by $r(n)$ is obtained as put a point on the beginning of $n$ like $r(34880)=0.34880$ and let $N_1=\Bbb N\cup\{0\}$ and similarly consider $\forall k\in N_1,$ $r_k: \Bbb N \to (0,1)$ by $r_k(n)=10^{-k}\cdot r(n)$.

Based on theorem $2$ we can define $f:\{(c,d)\,|\, (c,d)\subseteq [0.1,1)\}\to\Bbb N$ is a function that $f((c,d))$ is the least $n\in\Bbb N$ that $\exists t\in (c,d)\,\exists k\in\Bbb N$ that $p_n=t\cdot 10^k$ that $p_n$ is $n$_th prime and $\forall m\ge f((c,d))$ $\exists u\in (c,d)$ that $u\cdot 10^m\in\Bbb P$

and $g:(0,0.9)\cap (\bigcup _{k\in N_1} r_k(\Bbb N))\to\Bbb N,$ is a function by $\forall\epsilon\in (0,0.9)\cap (\bigcup _{k\in N_1} r_k(\Bbb N))$ $g(\epsilon)=max(\{f((c,d))\,|\, d-c=\epsilon,$ $(c,d)\subseteq [0.1,1)\})$. $$\\$$

Question $1$: Isn't $g$ an injective function? $$\\$$

Let $[a,a]:=\{a\}$

Question $2$: Assume $g$ isn't injection and $\forall n\in\Bbb N,\, h_n$ is the least subinterval of $[0.1,1)$ like $[a,b]$ in terms of size of $b-a$ such that $\{\epsilon\in (0,0.9)\cap (\bigcup _{k\in\Bbb N} r_k(\Bbb N))\,|\, g(\epsilon)=n\}\subsetneq h_n$ and it is clear $g(a)=n=g(b)$ now the question is $\forall n,m\in\Bbb N$ that $m\neq n$ is $h_n\cap h_m=\emptyset$?

$$\\$$ Thanks in advance.

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  • $\begingroup$ Do you have a reference for theorem 1? $\endgroup$ – Jack M Nov 13 '17 at 11:05
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    $\begingroup$ @JackM, these are a part of my theories and some mathematicians like @Wojowu and @reuns vouched theorem $1$ and said it is a result of prime number theorem! and of course clearly theorem $1$ is a result of theorem $2$ that it was proved by Adayah in this question. $\endgroup$ – user485602 Nov 15 '17 at 8:34
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    $\begingroup$ Why don't you read number theory books, where the exercices come with solutions.. $\endgroup$ – reuns Nov 18 '17 at 19:09
  • $\begingroup$ @reuns thank you, you are right. $\endgroup$ – user485602 Nov 20 '17 at 9:26
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  • 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}\}$

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