Analysis of the distribution of prime numbers. Part 1
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This part of the article demonstrates the author's functional and mathematical tools for the comparative analysis of certain power sequences, including a sequence of prime numbers. Particular attention is paid to identifying recurrently significant formula approximations for determining subsequent values in a sequence of prime numbers.
Story
Upper Paleolithic (c. 20,000 BC): The Ishango bone, found in Africa, contains nicks that may indicate an early understanding of prime numbers.
around 2000 BC: Egyptian mathematicians recorded their knowledge of prime numbers in the Rhind Papyrus, which contained problems involving factoring numbers.
VI-V centuries BC: Pythagoreans studied numbers, including prime numbers, and their properties, discovering their importance in various fields such as music and astronomy.
3rd century BC: Euclid, in the Elements, proves the infinity of prime numbers and introduces the concept of perfect numbers, related to Mersenne primes.
320 BC: Eratosthenes invents the Sieve of Eratosthenes, the first known algorithm for finding prime numbers.
1472: For the first time, Goldbach's hypothesis was put forward, suggesting that every even number greater than two can be expressed as the sum of two prime numbers.
1644: Maren Mersenne discovers Mersenne prime numbers for
And
.
1732: Leonhard Euler shows that the fifth power of Fermat's number is not a prime number, disproving Fermat's conjecture that all Fermat numbers are prime.
1750: Leonhard Euler discovers the 31st Mersenne prime number.
1852: Pafnuty Chebyshev proves Bertrand’s postulate, which states that between any number
and its double value
there is always a prime number.
1896: Jacques Hadamard and Charles Jean de la Vallée-Poussin prove a theorem about the distribution of prime numbers, known as the prime number theorem.
1996: The Great Internet Mersenne Prime Search (GIMPS) project begins, which aims to search for large Mersenne primes using distributed computing.
Practical use
Cryptography: Prime numbers are the basis for creating keys in encryption algorithms such as RSA. The complexity of the task of factoring large numbers consisting of two prime factors ensures the security of data on the Internet.
Internet security: Prime numbers are used to create secure Internet connections, where they serve as the basis for generating encryption keys and cryptographic protocols.
Radiology/Medical Imaging: In medical imaging, prime numbers can be used to improve the quality of images such as MRI or CT, allowing pathologies to be more accurately identified.
Mathematical research: Prime numbers have a rich field of study in mathematics with numerous applications and implications.
Error detection and correction: In telecommunications, prime numbers are used to generate error correction codes, which ensure accurate transmission of messages with automatic correction of possible errors.
These few examples may indicate the universality and importance of prime numbers in the modern world.
Entering tools
Some analytical apparatus will emerge for some determination of the complexities of certain sequences, with fundamental significance. All possible limit values are introduced, the number limited by the combinatorial laws of permutations, and significant sequences are given for each of the possible values, from a limit function of the form:
i/Lim x →0, x, ∞/ Lim n →∞ = Fδ,η = 0δ,ηXδ,η,∞δ,η
Here, x – is given arbitrarily and from a certain interval, as will be shown below in the corresponding and in elementary examples – as determinants, rushes to one of the given values,
n – Is an index-fixed value and or is independent of the values x:
i/ Lim x →0 / Lim n→∞ = F1.1 = 01
[1/1/1=105/05/2=050(3)/0(3)/3=03025/025/4=025…)[1/1/1=105/05/2=050(3)/0(3)/3=03025/025/4=025…)
i/ Lim x →x = 1 / Lim n→∞ = F1.2 = 02
[1/1/1=11/1/2=051/1/3=031/1/4=025…)[1/1/1=11/1/2=051/1/3=031/1/4=025…)
i/ Lim x →∞ / Lim n→∞ = F1.3 = 03
[1/1/1=11/2/2=0251/3/3=011/4/4=00625…)[1/1/1=11/2/2=0251/3/3=011/4/4=00625…)
Была выявлена первая группа нулевых значений F по δ, с приведением особых численно-последовательных примеров, также играющих роль в доказательстве некоторой возможности данных пределов. Как видно из числовых примеров, мы подбираем значения i таким образом, чтобы F с некоторого значения, стремилась к нулю, где x → 0 или x или ∞…
i/ Lim x →0 / Lim n→∞ = F2.1 = X4
[1/1/1=11/05/2=11/0(3)/3=11/025/4=1…)[1/1/1=11/05/2=11/0(3)/3=11/025/4=1…)
i/ Lim x → x = 1 / Lim n→∞ = F2.2 = X5
[1/1/1=12/1/2=13/1/3=14/1/4=1…)[1/1/1=12/1/2=13/1/3=14/1/4=1…)
i/ Lim x →∞ / Lim n→∞ = F2.2 = X6
[1/1/1=14/2/2=19/3/3=116/4/4=1…)[1/1/1=14/2/2=19/3/3=116/4/4=1…)
Была выявлена вторая группа ненулевых конечных F значений по δ.
i/ Lim x →0 / Lim n→∞ = F3.1 = ∞7
[1/1/1=11/025/2=21/0(1)/3=31/0625/4=4…)[1/1/1=11/025/2=21/0(1)/3=31/0625/4=4…)
i/ Lim x →x = 1 / Lim n→∞ = F3.2 = ∞8
[1/1/1=14/1/2=29/1/3=316/1/4=4…)[1/1/1=14/1/2=29/1/3=316/1/4=4…)
i/ Lim x →∞ / Lim n→∞ = F3.3 = ∞9
[1/1/1=18/2/2=227/3/3=364/4/4=4…)[1/1/1=18/2/2=227/3/3=364/4/4=4…)
Выявлена третья группа бесконечных значений F по δ.
From what has been revealed, it is useful to create the following table:
It can be seen that static i(Fδ,η) are located at 0203X4 and ∞7, and in other cases, i needs to be incremented to comply with the conditions of its limit. B 01i has to be incremented to zero, and in X5X6,∞8,∞9i is incremented to infinity…
…
For a better understanding of the “complexities” of certain sequences, it is worth introducing some reference values, compiled using the above table 0, for a number of natural numbers N, quadratic N², and prime numbers Π:
Let's start with the simple ones:
Let us determine which of the given functional limits the sequence of prime numbers belongs to. Since n is an indexically fixed value, it will be correlated and further analyzed when Pn= i.
where P is a prime number, and n determines the serial number of the prime number in the sequence, where the first prime number is 2 – i.e. 21
For a correct relationship, a theorem on the law of asymptotic distribution of prime numbers (LAP) is required – it is stated that for any arbitrary natural number n> 3, between the numbers n and 2n there is at least one prime number. And, index limit theorem on the limit of distribution of prime numbers (ILDP) – Pn/n =∞ , which is an obvious consequence of TZRP:
The distribution of prime numbers is correlated over three possible limits, of the form i = Пn:
The sequence of prime numbers for the first limit is correlated;
i = Pn/Lim x →0/Lim n →∞
A sequence of numbers that satisfies this limit takes on the following values:
[2/1/1=23/05/2=35/03/3=57/025/4=711/02/5=11…)[2/1/1=23/05/2=35/03/3=57/025/4=711/02/5=11…)
From here it is clear, according to Euclid’s theorem, that
i = Pn/Lim x →0/Lim n →∞ =∞7
The sequence of prime numbers for the second limit is correlated;
i = Pn/Lim x → x = 1/Lim n →∞
A sequence of numbers that satisfies this limit takes on the following values:
[2/1/1=23/1/2=155/1/3=1(6)7/1/4=17511/1/5=22…)[2/1/1=23/1/2=155/1/3=1(6)7/1/4=17511/1/5=22…)
hence from ILTP
i = Пn/Lim x → x = 1/Lim n →∞ = ∞8
The sequence of prime numbers for the last limit is correlated;
i = Pn/Lim x → ∞/Lim n →∞
A sequence of numbers that satisfies this limit takes on the following values:
[2/1/1=23/2/2=0755/3/3=057/4/4=0437511/5/5=0413/6/6=036117/7/7=03419/8/8=02923/9/9=02829/10/10=02931/11/11=025)[2/1/1=23/2/2=0755/3/3=057/4/4=0437511/5/5=0413/6/6=036117/7/7=03419/8/8=02923/9/9=02829/10/10=02931/11/11=025…)
From TZRP, it follows that
i = Пn/Lim x → ∞/Lim n →∞ = 03
For the obtained limit values in the form i = Пn Let’s also make a table and highlight the obtained values with a dash:
In the same way, we will compile tables for natural and quadratic distributions of numbers:
Thus, from a comparison of the table ratios it becomes completely clear that;
nΨ>1<2 = Πn
Indeed, this can be seen by looking at the following values;
Let us now compile a limit table for the sequence of numbers N1.5
It can be seen that the limiting value of the distribution of numbers N1.5the distribution of prime numbers is similar to the limit values… By comparing with the values in Table 4, we can clearly come to the conclusion that;
nΨ>1<1.5 = Πn
From the author:
7 years of research in prime number theory.
Self-taught.
21 century.
Expect the next parts
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