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[17] viXra:2207.0177 [pdf] replaced on 2022-09-19 19:51:09
Authors: Bassam Abdul-Baki
Comments: 17 Pages.
Some interesting properties arise when value-counting the integers sequentially up to N using N digits or fingers and comparing the number of values to the prime-exact equation; with a simple method for testing primes and prime powers (particularly Mersenne and Fermat primes).
Category: Number Theory
[16] viXra:2207.0125 [pdf] submitted on 2022-07-20 01:36:47
Authors: Tianshu Zhang
Comments: 17 Pages.
The subject of this article is exactly to analyze and prove Beal’s Conjecture. First, classify A, B and C according to their respective parity, and two types of AX+BY≠CZ are excluded, for they have nothing to do with the conjecture. Next, several types of AX+BY=CZ under the necessary constraints are exemplified, where A, B, and C have at least one common prime factor. Secondly, divide AX+BY≠CZ under the necessary constraints into four inequalities under the known constraints, in order to make more detailed proofs, where A, B and C have not any common prime factor. Then, expound the interrelation between an even number as the center of symmetry and a sum of two odd numbers in the symmetry, and draw four conclusions which can be used as basis for judging certain results in the processes of proofs for the four inequalities. After that, two inequalities under the known constraints are proved by the mathematical induction. Then again, two other inequalities under the known constraints are proved by the reduction to absurdity. Finally, after comparing AX+BY=CZ and AX+BY≠CZ under necessary constraints, the conclusion is that the Beal's conjecture is true.
Category: Number Theory
[15] viXra:2207.0119 [pdf] submitted on 2022-07-19 01:43:32
Authors: Theophilus Agama
Comments: 5 Pages. This paper improves on the bounds in previous works and it is more explicit.
In this paper we prove that there exists an addition chain producing 2^n-1 of length delta(2^n-1) satisfying the inequality delta(2^n-1)leq 2n-1-2left lfloor frac{n-1}{2^{lfloor frac{log n}{log 2}floor}}ight floor+lfloor frac{log n}{log 2}flooronumber where lfloor cdot floor denotes the floor function.
Category: Number Theory
[14] viXra:2207.0111 [pdf] replaced on 2024-07-16 19:53:52
Authors: Tianshu Zhang
Comments: 18 Pages.
In this article, we classify gradually positive integers ≥2, and express each and every class of positive integers into a sum of 3 unit fractions. First, divide all positive integers ≥2 into 8 kinds, and then formulate each of 7 kinds of these 8 kinds into a sum of 3 unit fractions. For the unsolved kind, divide it into 3 genera, and then formulate each of 2 genera of these 3 genera into a sum of 3 unit fractions. For the unsolved genus, further divide it into 5 sorts, and formulate each of 3 sorts of these 5 sorts into a sum of 3 unit fractions. For two unsolved sorts, let each of them be expressed as a sum of an unit fraction plus a true fraction, and that take out the unit fraction as one of 3 unit fractions which express the sort as the sum. After that, if the true fraction can be transformed identically into an unit fraction, then we follow the formula that Ernst G. Straus made to transform either of these two unit fractions into a sum of two each other’s- distinct unit fractions, such that this part of the unsolved sort becomes a sum of 3 unit fractions. If the true fraction can not be transformed identically into an unit fraction, then we let it to equal the sum of an unit fraction plus another true fraction, and that take out the unit fraction as one of 3 unit fractions which express the sort as the sum. Next, prove that another proper fraction can be identically converted into an unit fraction. Due to c≥0, above two cases exist surely when c is taken different values.
Category: Number Theory
[13] viXra:2207.0086 [pdf] submitted on 2022-07-12 20:59:39
Authors: Krishna Srinivasan, Rajaram Gana
Comments: 9 Pages.
A multidimensional number will not be viewed as a single real scalar value, rather,as a set of scalar values, each associated with a dimension. This gives rise to variations of"complex numbers", and consequently, Euler’s formula. The properties of complex numbers,such as the product of magnitudes being equal to the magnitude of the products, may also beapplicable in the case of multidimensional numbers, depending on how they are constructed.Although similar ideas exist, such as a hypercomplex number, the differences will be discussed.
Category: Number Theory
[12] viXra:2207.0072 [pdf] submitted on 2022-07-09 06:55:27
Authors: Lushi Liu
Comments: 10 Pages.
If the solution set of two equations(group) has the same algebraic operation structure based on equations (group), the two equations (group) are called isomorphic equations (group), and the solution set of these two equations (group) is the equivalent solution set.
Category: Number Theory
[11] viXra:2207.0061 [pdf] submitted on 2022-07-09 02:58:04
Authors: Fortuné Alain Junior Backoulas
Comments: 12 Pages.
This small excerpt from our paper on the Collatz conjecture is intended to give new directions to those who are working on the understanding of this problem, but, above all, for those who have invested in the solution of this problem. We have limited ourselves in this document to raising awareness of the approach we have adopted, which was certainly not the right one. This is what makes such a simple problem seem so difficult to solve. As this man said: it is difficult to paint the real landscape when you accidentally end up on another landscape. In this paper, we make a contribution to the understanding that the trivial cycle of basis 1 2 4 is only one case among a whole set of trivial cycles. And that in this generality, the trivial cycle of Collatz has the same properties as the other cycles and that it should not always be taken as a basic element. In this sense, trying to understand it does not bring us back to determining the behavior of this cycle because, other notions must be understood beforehand. Hence; in our opinion, all the difficulties that mathematicians and others encounter today and since.
Category: Number Theory
[10] viXra:2207.0060 [pdf] replaced on 2022-09-04 06:38:39
Authors: Hyeon Jun Ahn
Comments: 9 Pages. This paper is a etude
Let there exists m > 0 such that gn = O((logpn)m), then∀k > 0, ∃M ∈ N s.t. n ≥ M ⇒ gn := pn+1 − pn < pknwhere pn is nth prime number, O is big O notation, log is natural logarithm.This lead to a corollary for Andrica conjecture, Oppermann conjecture.
Category: Number Theory
[9] viXra:2207.0058 [pdf] submitted on 2022-07-07 23:33:40
Authors: James Edwin Rock
Comments: 4 Pages.
Let Pn be the n-th prime. For twin primes Pn u2013 Pn-1 = 2. Let X be the number of (6j u20131, 6j+1) pairs in the open interval [Pn, Pn2 ]. The actual number of twin primes TPAn in [Pn, Pn2 ] is ((Pn - an) /Pn)((Pn-1 - an-1) /Pn-1)((Pn-2 - an-2)/Pn-2)u2026((5 - a3) /5)(X). P3=5, 1.7 < an,an-1,u2026,a3 less than 2.3. We exhibit a formula showing as Pn increases, the actual number of twin primes TPAn in the interval [Pn, Pn2 ] also increases. Let Pn u2013 Pn-1 = c. For n ≥ 4, (TPAn-1)(1+(2c u20132)/2Pn-1+(c2u20132c)/2Pn-12) less than TPAn.
Category: Number Theory
[8] viXra:2207.0045 [pdf] submitted on 2022-07-05 22:56:54
Authors: Edgar Valdebenito
Comments: 4 Pages.
Bungus equation and Pi.
Category: Number Theory
[7] viXra:2207.0027 [pdf] replaced on 2022-07-12 13:26:15
Authors: Barchino, R.; Segura, J. J.
Comments: 19 Pages.
This paper provides a study on composite numbers and delivers 5 equations that, when put together, generate all existing composite numbers. This is done by sorting all natural numbers in 6 groups and following the mathematical reasoning that explains the generation of all composite numbers for each one of these groups. Derived from this, two different ways to obtain prime numbers by iteration are provided, although slower in computational speed than the ones existing today. An iteration method to find twin prime numbers is also described.
Category: Number Theory
[6] viXra:2207.0026 [pdf] submitted on 2022-07-04 22:35:09
Authors: Edgar Valdebenito
Comments: 5 Pages.
We give some formulas for the Grothendieck-Krivine number.
Category: Number Theory
[5] viXra:2207.0021 [pdf] submitted on 2022-07-03 22:27:25
Authors: Yuji Masuda
Comments: 1 Page.
The purpose of this short paper is to spread the interest and fascination of mathematics. I have also written this article again with the aim of making mathematics more interesting to my readers.
Category: Number Theory
[4] viXra:2207.0016 [pdf] submitted on 2022-07-02 07:15:14
Authors: Ramaswamy Krishnan
Comments: 9 Pages.
The abstract of the paper is included in the paper itself as Synopsis.
Category: Number Theory
[3] viXra:2207.0013 [pdf] replaced on 2024-07-28 23:12:42
Authors: Abdelmajid Ben Hadj Salem
Comments: 102 Pages. Many modifications are added since the last version (December 2022).
This monograph presents the proofs of 4 important conjectures in the field of the number theory, namely:- Beal's conjecture. - The Riemann Hypothesis.- The c smaller than R^{1.63} conjecture is true. - The abc conjecture is true. We give the details of the different proofs.
Category: Number Theory
[2] viXra:2207.0012 [pdf] submitted on 2022-07-03 01:51:06
Authors: Daoudi Rédoane
Comments: 2 Pages.
Here I present a prime-generating sequence based on the Wilson's theorem.
Category: Number Theory
[1] viXra:2207.0011 [pdf] replaced on 2026-04-13 06:54:18
Authors: Denis Gallet
Comments: 11 Pages. In this update, I write the correct proof for Log G(1/5),Log G(1/8) and Log G(1/12)
I have already published several months ago in the papers "Values of Barnes Function" and "Another Values of Barnes Function and Formulas" in total 16 conjectural formulas that I find with unsualmethods.So, in this article, I write the proof of 16 formulas.
Category: Number Theory
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