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[21] viXra:2205.0155 [pdf] submitted on 2022-05-31 20:35:31
Authors: Takamasa Noguchi
Comments: 10 Pages.
For { p-1 = q^L*m ( ∤ q^x ∨ | q^x (x larger than L))}, it is the deterministic algorithm. The previously created calculation method was for a single prime number, but a method to calculate multiple primes has been added. The original calculation method has also been partially modified.
To find the nth root, we need to factoer n into prime factors. In some case, primitive roots are needed. If you don't know these, use the Tonelli-Shanks algorithm.
Category: Number Theory
[20] viXra:2205.0151 [pdf] submitted on 2022-05-31 08:32:56
Authors: Theophilus Agama
Comments: 7 Pages.
In this paper we prove that there infinitely many cousin primes by deducing the lower bound
\begin{align}
\sum \limits_{\substack{p\leq x\\p,p+4\in \mathbb{P}\setminus \{2\}}}1\geq (1+o(1))\frac{x}{2\mathcal{C}\log^2 x}\nonumber
\end{align}where $\mathcal{C}:=\mathcal{C}(4)>0$ fixed and $\mathbb{P}$ is the set of all prime numbers. In particular it follows that
\begin{align}
\sum \limits_{p,p+4\in \mathbb{P}\setminus \{2\}}1=\infty\nonumber
\end{align}by taking $x\longrightarrow \infty$ on both sides of the inequality. We start by developing a general method for estimating correlations of the form
\begin{align}
\sum \limits_{n\leq x}G(n)G(n+l)\nonumber
\end{align}for a fixed $1\leq l\leq x$ and where $G:\mathbb{N}\longrightarrow \mathbb{R}^{+}$.
Category: Number Theory
[19] viXra:2205.0144 [pdf] replaced on 2024-07-04 21:29:25
Authors: Shan Jian Wang
Comments: 7 Pages.
Conjecture: Any even number greater than 2 can be written as the sum of two prime numbers. Does the prime pair exist universally? If does, is the prime pair unique relatively? If not, how many prime pairs are there in an even? Method: Triangular lattice Result: The number of prime pairs in an even can be expressed analytically and graphically Conclusion: Any even number greater than 2 can be written as the sum of two prime numbers.
Category: Number Theory
[18] viXra:2205.0132 [pdf] replaced on 2022-06-04 23:50:04
Authors: Marco Rolando Burgos Chambi
Comments: 29 Pages.
This paper will prove that the Riemann Hypothesis is true., based on the following statements:
-The resulting value of the Euler-Riemann zeta function ζ(k) is the center of a spiral on the complex plane, where k ∈ C.
-The center of this spiral when ζ(k) = 0, coincides with the origin of coordinates of the complex plane.
-There exists a function related to this spiral, obtained from Bernoulli's sum of powers, which allows to calculate the zeta funtion.
Category: Number Theory
[17] viXra:2205.0129 [pdf] submitted on 2022-05-25 17:51:44
Authors: Stephen Marshall
Comments: 9 Pages.
The Collatz conjecture is a conjecture in mathematics that concerns a sequence defined as follows: start with any positive integer n. Then each term is obtained from the previous term as follows: if the previous term is even, the next term is one half of the previous term. If the previous term is odd, the next term is 3 times the previous term plus 1. The conjecture is that no matter what value of n, the sequence will always reach 1. For example, starting with n = 12, one gets the sequence 12, 6, 3, 10, 5, 16, 8, 4, 2, 1.
As of 2020, the conjecture has been checked by computer for all starting values up to 268 ≈ 2.95×1020. The eccentric Hungarian mathematician Paul Erdős claimed that "Mathematics is not yet ready for such problems," and referred to the conjecture as "Hopeless. Absolutely hopeless."
The Collatz Conjecture describes the iterations of integers applied to a very simple function. The conjecture specifically states: "Starting from any positive integer n, iterations of the function C(n) will eventually reach the number 1. Thereafter iterations will cycle taking successive values 1, 4, 2, 1, 4, 2, 1 ..." (Lagarias, 2010).
To define a basic term, an integer n will be defined as odd when n ≡ 1 (mod 2). Likewise, n will be defined as even when n ≡ 0 (mod 2). With those common terms specified, the following is the function known as the Collatz function:
3n+1 if n is odd
C(n) =
n/2 if n is even
The Collatz function is named as such with respect to its originator.
The Collatz conjecture was made in 1937 by Lothar Collatz. Again, as of 2020, the conjecture has been checked by computer for all starting values up to 268 ≈ 2.95×1020, but very little progress has been made toward proving the conjecture. The author is shocked that such a simple proof exists. The author is humbly grateful for this first proof as well, as it came to me in a “flash” in such a way as I believe it was given to me (my brothers Ben and Phil will understand this). The second proof did not come to me via a “flash” experience, as the first one was.
Category: Number Theory
[16] viXra:2205.0106 [pdf] replaced on 2023-01-13 13:02:00
Authors: Theophilus Agama
Comments: 6 Pages. The requirements in the diagonal method has been greatly simplified
In this paper we introduce and develop the method of diagonalization of functions $f:mathbb{N}longrightarrow mathbb{R}$. We apply this method to show that the equations of the form $Gamma_r(n)+k=m^2$ has a finite number of solutions $nin mathbb{N}$ with $n>r$ for any fixed $k,rin mathbb{N}$, where $Gamma_r(n)=n(n-1)cdots (n-r)$ denotes the $r^{th}$ truncated Gamma function.
Category: Number Theory
[15] viXra:2205.0096 [pdf] replaced on 2022-06-13 20:42:37
Authors: Huang Shan
Comments: 2 Pages.
This paper presents a method for drawing an image of the distribution of prime numbers in natural numbers.
Category: Number Theory
[14] viXra:2205.0084 [pdf] submitted on 2022-05-16 16:19:13
Authors: Theophilus Agama
Comments: 5 Pages.
Let $\delta(n)$ denotes the length of an addition chain producing $n$. In this paper we prove that the exists an addition chain producing $2^n-1$ whose length satisfies the inequality $$\delta(2^n-1)\lesssim n-1+\iota(n)+\frac{n}{\log n}+1.3\log n\int \limits_{2}^{\frac{n-1}{2}}\frac{dt}{\log^3t}+\xi(n)$$ where $\xi:\mathbb{N}\longrightarrow \mathbb{R}$. As a consequence, we obtain the inequality $$\iota(2^n-1)\lesssim n-1+\iota(n)+\frac{n}{\log n}+1.3\log n\int \limits_{2}^{\frac{n-1}{2}}\frac{dt}{\log^3t}+\xi(n)$$ where $\iota(n)$ denotes the length of the shortest addition chains producing $n$.
Category: Number Theory
[13] viXra:2205.0078 [pdf] submitted on 2022-05-14 14:09:08
Authors: Junho Choi
Comments: 5 Pages.
I found an alternative form of Hardy-Littlewood Conjecture using a corollary of Mertens’ 2nd theorem. This new form would be more useful since it has a theoretical background and is more likely to be proved.
Category: Number Theory
[12] viXra:2205.0075 [pdf] replaced on 2024-11-30 23:13:02
Authors: Mar Detic
Comments: 4 Pages.
This paper provides a proof of primality using a set construction S. Specifically, it demonstrates that a positive integer p > 1 is prime if and only if p /∈ S. The set S is defined in terms of prime divisors of p and is constructed by considering natural numbers within certain bounds. Examples are provided to illustrate the application of thiscriterion for both prime and composite numbers, including p = 121.
Category: Number Theory
[11] viXra:2205.0074 [pdf] replaced on 2022-07-03 01:19:43
Authors: E. Zhou, M. Zhou
Comments: 2 Pages.
ABSTRACT. Here, we provide a novel set of expressions for zeta (3) which are systematic results.
Category: Number Theory
[10] viXra:2205.0071 [pdf] submitted on 2022-05-12 04:43:47
Authors: Marko V. Jankovic
Comments: 6 Pages.
In this paper it is going to be proved that weak Goldbach's conjecture can not hold. The proof is based on fundamental theorem of arithmetic. This paper presents the idea that is simple modification (extension) of the idea used to prove that strong Goldbach's conjecture cannot hold.
Category: Number Theory
[9] viXra:2205.0070 [pdf] submitted on 2022-05-12 04:45:53
Authors: Marko V. Jankovic
Comments: 6 Pages.
In this paper it is going to be proved that Schnilermann's theorem can not hold. A disproof is based onfundamental theorem of arithmetic. However, since the proof of the theorem is widely accepted, that creates an unusual paradox that theorem can be proved and disproved at the same time.
Category: Number Theory
[8] viXra:2205.0069 [pdf] submitted on 2022-05-12 23:46:19
Authors: Huang Shan
Comments: 4 Pages. (Note by viXra Admin: Pseudonym is not permitted, and there shouldn't be a hyphen between first name and last name)
If the Goldbach conjecture can be represented by a set, then the Goldbach conjecture is only
one of the common mapping relations represented by a certain set in a wider range. In order to prove
that I'm not kidding, I'll write the method of prime distribution image in the first paragraph of the article.
This is easy to verify. You can know it's true by drawing it.
Category: Number Theory
[7] viXra:2205.0064 [pdf] submitted on 2022-05-11 23:25:45
Authors: Usama Shamsuddin Thakur
Comments: 7 Pages.
Chaotic number system is a tool which was needed to solve the mysteries of today’s modern
mathematical problems the unsolved millennium prize problems namely 1. Riemann Hypothesis,
2. Navier Stokes Equations, 3. Yang Mills Theory and Mass Gap. These are some of the names
of problems which we will see in this paper can be solved I don’t claim that I have solved these
problems but certainly I have given a new perspective on them. We use chaotic numbers and
axiomatic properties of these numbers which is denoted by ���� and is a super set of complex
numbers. The numbers are dynamic and always changing the operations on them are different and
they match the exact description of quantum mechanics as the root of inventing these numbers
was to understand quantum particles and having a clear understanding by the mathematics that
works with the same uncertainty.
Category: Number Theory
[6] viXra:2205.0046 [pdf] submitted on 2022-05-08 08:05:59
Authors: Arthur V. Shevenyonov
Comments: 3 Pages. primality/decomposition routines rendered minimalist
Based on the ubiquitous ‘floor’ function, a constructive primality test (tantamount to a dual de- & re-composition algorithm) is proposed alongside auxiliary conjectures. Implications potentially border on domains as co-distant and versatile as, the #-scores, Shapley value, and ABC conjecture to name but a few.
Category: Number Theory
[5] viXra:2205.0041 [pdf] replaced on 2023-01-30 01:40:34
Authors: Hajime Mashima
Comments: 22 Pages.
In this paper, the symmetry is shown for two and three pairs of rings.
Category: Number Theory
[4] viXra:2205.0028 [pdf] submitted on 2022-05-05 11:51:55
Authors: Theophilus Agama
Comments: 4 Pages.
In this paper we extend the so-called notion of addition chains and prove an analogue of Scholz's conjecture on this chain. In particular, we obtain the inequality $$\iota^{\lfloor \frac{n-1}{2}\rfloor}(2^n-1)\leq n+\iota(n)$$ where $\iota(n)$ and $\iota^{\lfloor \frac{n-1}{2}\rfloor}(n)$ denotes the length of the shortest addition chain and the shortest addition chain of degree $\lfloor \frac{n-1}{2}\rfloor$, respectively, producing $n$.
Category: Number Theory
[3] viXra:2205.0027 [pdf] submitted on 2022-05-05 20:31:22
Authors: Ryujin Choe
Comments: 21 Pages. (Note by viXra Admin: Powerpoint form is not acceptable form - Please replace and do not use in the future)
Proof of Twinprime conjecture , Goldbach's conjecture.
Category: Number Theory
[2] viXra:2205.0024 [pdf] submitted on 2022-05-04 04:38:34
Authors: Marko V. Jankovic
Comments: 3 Pages.
In this paper it is going to be proved that strong Goldbach's conjecture can not hold. The proof is based on fundamental theorem of arithmetic.
Category: Number Theory
[1] viXra:2205.0008 [pdf] submitted on 2022-05-02 12:11:12
Authors: Lucian M Ionescu
Comments: 3 Pages. Preliminary version based on the presentation available at https://about.illinoisstate.edu/lmiones/research/
Prime numbers have a rich structure, when viewed as sizes of finite fields. Iteration of an analysis as Klein geometry yields their deconstruction into simpler primes: the POSet structure.
Reversing the process is Euclid's trick of generating new primes.
A generalization of this is used by McCanney to cover the set of primes away from primorials as centers. This fast algorithm has a ``propagation'' flavor.
Generating primes in this manner is also related with Goldbach's Conjecture.
Category: Number Theory
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