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== Goldbach's conjecture ==
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=== Goldbach's conjecture ===
  
 
'''Main theorem''': Let $\mathbb{P}$ is the set prime numbers and $S$ is a set that has been made as below: put a point at 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.
 
'''Main theorem''': Let $\mathbb{P}$ is the set prime numbers and $S$ is a set that has been made as below: put a point at 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.
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  Alireza Badali 22:21, 8 May 2017 (CEST)
 
  Alireza Badali 22:21, 8 May 2017 (CEST)
  
== Polignac's conjecture ==
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=== Polignac's conjecture ===
  
 
In number theory, Polignac's conjecture was made by Alphonse de Polignac in 1849 and states: For any positive even number $n$, there are infinitely many prime gaps of size $n$. In other words: There are infinitely many cases of two consecutive prime numbers with difference $n$. (Tattersall, J.J. (2005), Elementary number theory in nine chapters, Cambridge University Press, ISBN: 978-0-521-85014-8, p. 112) Although the conjecture has not yet been proven or disproven for any given value of n, in 2013 an important breakthrough was made by Zhang Yitang who proved that there are infinitely many prime gaps of size n for some value of n < 70,000,000.(Zhang, Yitang (2014). "Bounded gaps between primes". Annals of Mathematics. 179 (3): 1121–1174. MR 3171761. Zbl 1290.11128. doi:10.4007/annals.2014.179.3.7. _  Klarreich, Erica (19 May 2013). "Unheralded Mathematician Bridges the Prime Gap". Simons Science News. Retrieved 21 May 2013.) Later that year, James Maynard announced a related breakthrough which proved that there are infinitely many prime gaps of some size less than or equal to 600.(Augereau, Benjamin (15 January 2013). “An old mathematical puzzle soon to be unraveled? Phys.org. Retrieved 10 February 2013.)
 
In number theory, Polignac's conjecture was made by Alphonse de Polignac in 1849 and states: For any positive even number $n$, there are infinitely many prime gaps of size $n$. In other words: There are infinitely many cases of two consecutive prime numbers with difference $n$. (Tattersall, J.J. (2005), Elementary number theory in nine chapters, Cambridge University Press, ISBN: 978-0-521-85014-8, p. 112) Although the conjecture has not yet been proven or disproven for any given value of n, in 2013 an important breakthrough was made by Zhang Yitang who proved that there are infinitely many prime gaps of size n for some value of n < 70,000,000.(Zhang, Yitang (2014). "Bounded gaps between primes". Annals of Mathematics. 179 (3): 1121–1174. MR 3171761. Zbl 1290.11128. doi:10.4007/annals.2014.179.3.7. _  Klarreich, Erica (19 May 2013). "Unheralded Mathematician Bridges the Prime Gap". Simons Science News. Retrieved 21 May 2013.) Later that year, James Maynard announced a related breakthrough which proved that there are infinitely many prime gaps of some size less than or equal to 600.(Augereau, Benjamin (15 January 2013). “An old mathematical puzzle soon to be unraveled? Phys.org. Retrieved 10 February 2013.)
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  Alireza Badali 13:17, 21 August 2017 (CEST)
 
  Alireza Badali 13:17, 21 August 2017 (CEST)
  
== Landau's forth problem ==
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=== Landau's forth problem ===
  
 
Landau's forth problem: Are there infinitely many primes $p$ such that $p−1$ is a perfect square? In other words: Are there infinitely
 
Landau's forth problem: Are there infinitely many primes $p$ such that $p−1$ is a perfect square? In other words: Are there infinitely
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  Alireza Badali 20:47, 21 September 2017 (CEST)
 
  Alireza Badali 20:47, 21 September 2017 (CEST)
  
== Grimm's conjecture ==
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=== Grimm's conjecture ===
  
 
In mathematics, and in particular number theory, Grimm's conjecture (named after Karl Albert Grimm) states that to each element of a set of consecutive composite numbers one can assign a distinct prime that divides it. It was first published in American Mathematical Monthly, 76(1969) 1126-1128.
 
In mathematics, and in particular number theory, Grimm's conjecture (named after Karl Albert Grimm) states that to each element of a set of consecutive composite numbers one can assign a distinct prime that divides it. It was first published in American Mathematical Monthly, 76(1969) 1126-1128.
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  Alireza Badali 23:30, 20 September 2017 (CEST)
 
  Alireza Badali 23:30, 20 September 2017 (CEST)
  
== Lemoine's conjecture ==
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=== Lemoine's conjecture ===
  
 
In number theory, Lemoine's conjecture, named after Émile Lemoine, also known as Levy's conjecture, after Hyman Levy, states that all odd integers greater than $5$ can be represented as the sum of an odd prime number and an even semiprime.
 
In number theory, Lemoine's conjecture, named after Émile Lemoine, also known as Levy's conjecture, after Hyman Levy, states that all odd integers greater than $5$ can be represented as the sum of an odd prime number and an even semiprime.
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  Alireza Badali 00:30, 27 September 2017 (CEST)
 
  Alireza Badali 00:30, 27 September 2017 (CEST)
  
== About analytic number theory ==
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=== About analytic number theory ===
  
 
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem).
 
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem).
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:Thank you. Alireza Badali 16:00, 6 December 2017 (CET)
 
:Thank you. Alireza Badali 16:00, 6 December 2017 (CET)
  
==Question==
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===Question===
  
 
You often mention "Formula of prime numbers". What do you mean? This is not a well-defined mathematical object, but a vague idea, with a lot of ''non-equivalent'' interpretations. See for instance [https://en.wikipedia.org/wiki/Formula_for_primes], [http://mathworld.wolfram.com/PrimeFormulas.html], [https://www.quora.com/What-is-the-formula-of-prime-numbers], [https://primes.utm.edu/notes/faq/p_n.html], [https://math.stackexchange.com/questions/1257/is-there-a-known-mathematical-equation-to-find-the-nth-prime], [https://thatsmaths.com/2016/06/09/prime-generating-formulae/] etc. [[User:Passer By|Passer By]] ([[User talk:Passer By|talk]]) 19:24, 4 December 2017 (CET)
 
You often mention "Formula of prime numbers". What do you mean? This is not a well-defined mathematical object, but a vague idea, with a lot of ''non-equivalent'' interpretations. See for instance [https://en.wikipedia.org/wiki/Formula_for_primes], [http://mathworld.wolfram.com/PrimeFormulas.html], [https://www.quora.com/What-is-the-formula-of-prime-numbers], [https://primes.utm.edu/notes/faq/p_n.html], [https://math.stackexchange.com/questions/1257/is-there-a-known-mathematical-equation-to-find-the-nth-prime], [https://thatsmaths.com/2016/06/09/prime-generating-formulae/] etc. [[User:Passer By|Passer By]] ([[User talk:Passer By|talk]]) 19:24, 4 December 2017 (CET)
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:::::::Okay, thank you for your attention to my page. Alireza Badali 20:13, 7 December 2017 (CET)
 
:::::::Okay, thank you for your attention to my page. Alireza Badali 20:13, 7 December 2017 (CET)
  
== From the ''Main theorem'' ==
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=== From the ''Main theorem'' ===
  
 
Assume $H$ is a mapping from $(0.1,1)$ on to $(0.1,1)$ given by $H(x)=(10x)^{-1}$, and let $T=H(S)$, $T$ is an interesting set for its members because a member of $S$ like $0.a_1a_2a_3...a_n$ that $a_j$ is $j$-th digit for $j=1,2,3, ... ,n$ is basically different with ${a_1.a_2a_3a_4...a_n}^{-1}$ in $T$.
 
Assume $H$ is a mapping from $(0.1,1)$ on to $(0.1,1)$ given by $H(x)=(10x)^{-1}$, and let $T=H(S)$, $T$ is an interesting set for its members because a member of $S$ like $0.a_1a_2a_3...a_n$ that $a_j$ is $j$-th digit for $j=1,2,3, ... ,n$ is basically different with ${a_1.a_2a_3a_4...a_n}^{-1}$ in $T$.
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  Alireza Badali 15:59, 30 March 2018 (CEST)
 
  Alireza Badali 15:59, 30 March 2018 (CEST)
  
==Other problems==
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===Other problems===
  
 
$1)$ A basic system like decimal system must be existed as an axiom and another systems must be defined based on it, otherwise concept of numbers is obscure. _ Summation of digits of a natural number with consideration number of zeros in middle of number and not in its beginning (with partition by number of zeros) is related to divisors of that number particularly for prime numbers.
 
$1)$ A basic system like decimal system must be existed as an axiom and another systems must be defined based on it, otherwise concept of numbers is obscure. _ Summation of digits of a natural number with consideration number of zeros in middle of number and not in its beginning (with partition by number of zeros) is related to divisors of that number particularly for prime numbers.
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  Alireza Badali 15:59, 30 March 2018 (CEST)
 
  Alireza Badali 15:59, 30 March 2018 (CEST)
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== A theory ==
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Alireza Badali 08:26, 31 March 2018 (CEST)

Revision as of 06:26, 31 March 2018

My whole notes

A theory

Alireza Badali 08:26, 31 March 2018 (CEST)
How to Cite This Entry:
Musictheory2math. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Musictheory2math&oldid=43042