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[18] viXra:2107.0153 [pdf] replaced on 2023-11-12 23:56:35
Authors: Tae Beom Lee
Comments: 12 Pages.
Twin Prime Conjecture(TPC) states that there are infinitely many prime pairs (p, p + 2), where p is prime. But, up to date there is no valid proof of TPC. To prove TPC we devised Twin Prime Model Table(TPMT) and Sieve Functions(SFs). TPMT is a 2-dimensional table representation of all possible twin prime pairs(TPPs). SF is a sine function, f_i (x)=sin πx/p_i , where pi is the ith prime. SFs functionally represent the sieve of Eratosthenes because all zeros of f_i (x) can’t be prime exept the first zero. TPMT explicitly shows the mechanism of how TPPs are found from the possible twin prime pairs. To functionally represent this mechanism we introduced various sinusoidal functions. And by using properties of sinusoidal functions we proved TPC.
Category: Number Theory
[17] viXra:2107.0147 [pdf] submitted on 2021-07-24 20:41:30
Authors: Miguel Cerdá Bennassar
Comments: 3 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]
we present in this paper a generating function of periodic sequences with eligible cycle.
Category: Number Theory
[16] viXra:2107.0137 [pdf] submitted on 2021-07-23 19:04:08
Authors: Dmitri Martila
Comments: 6 Pages.
In this short note, I provide a proof for the Riemann Hypothesis. You are free not to get enlightened about that fact. But please pay respect to new dispositions of the Riemann Hypothesis and research methods in this note. I start with Dr.Zhu who was the first to show me that instead of the known 40 %, the maximum percentage of the zeroes of the Riemann zeta function belongs to the 1/2 critical line.
Category: Number Theory
[15] viXra:2107.0136 [pdf] submitted on 2021-07-23 07:27:20
Authors: Konstantinos Smpokos
Comments: 9 Pages. Published in the Bulletin of the Hellenic Mathematical Society.
In this article we will examine the behavior of certain free abelian subgroups of the multiplicative group of the positive rationals and their relationship with the group of units of integers modulo $n$.
Category: Number Theory
[14] viXra:2107.0135 [pdf] submitted on 2021-07-23 07:48:45
Authors: Konstantinos Smpokos
Comments: 11 Pages.
Lehmer's totient problem asks if there exists a composite number $d$ such that its totient divide $d-1$. In this article we generalize the Lehmer's totient problem in algebraic number fields. We introduce the notion of a Lehmer number. Lehmer numbers are defined to be the natural numbers which obey the Lehmer's problem in the ring of algebraic integers of a number field.
Category: Number Theory
[13] viXra:2107.0121 [pdf] replaced on 2021-08-11 20:02:25
Authors: Theophilus Agama
Comments: 8 Pages. A revision file
In this paper we develop the method of circle of partitions and associated statistics. As an application we prove conditionally the binary Goldbach conjecture. We develop series of steps to prove the binary Goldbach conjecture in full. We end the paper by proving the binary Goldbach conjecture for all even numbers exploiting the strategies outlined.
Category: Number Theory
[12] viXra:2107.0120 [pdf] submitted on 2021-07-19 21:31:40
Authors: Tae Tae Beom
Comments: 11 Pages.
Fermat's Last Theorem(FLT) states that there is no positive integer set (a,b,c,n) which satisfies a^n+b^n=c^n when n≥3. In this thesis, we related FLT to two polynomial equations. By doing so, we could analyze whether those two equations have equivalence properties in four aspects, ①irreducible factoring equivalence, ②constant term equivalence, ③ rational root factor equivalence and ④ odd-even property equivalence of a,b,c. What we found is that those two equations can not have equivalence properties in all four aspects which is enough to prove FLT.
Category: Number Theory
[11] viXra:2107.0117 [pdf] submitted on 2021-07-20 10:08:52
Authors: Juan Elias Millas Vera
Comments: 2 Pages.
In this paper we are going to see two theoretical expressions in reference of canonical representations. Based in the classic definition of positive integers we can use some mathematical tools to define the subsets of composite and prime numbers in their canonical form.
Category: Number Theory
[10] viXra:2107.0112 [pdf] submitted on 2021-07-19 20:10:17
Authors: Antoni Cuenca
Comments: 17 Pages.
Abstract. We introduce a topology in the set of natural numbers via a subbase of open sets. With this topology, we obtain an irreducible
(hyperconnected) space with no generic points. This fact allows proving that the cofinite intersections of subbasic open sets are always empty. That implies the validity of the Twin Prime Conjecture. On the other hand, the existence of strictly increasing chains of subbasic open sets shows that the Polignac Conjecture is false for an infinity of cases.
Category: Number Theory
[9] viXra:2107.0105 [pdf] submitted on 2021-07-18 16:49:39
Authors: Suaib Lateef
Comments: 6 pages
In this paper, we present an identity involving Tribonacci Numbers. We will prove this identity by extending the number of variables of Candido's identity to three.
Category: Number Theory
[8] viXra:2107.0095 [pdf] submitted on 2021-07-15 20:39:35
Authors: Bertrand Wong
Comments: 15 Pages. Published in an international mathematics journal. Thoroughly edited & re-arranged by the Editor-in-Chief of the journal.
The primes, including the twin primes and the other prime pairs, are the building-blocks
of the integers. Euclid’s proof of the infinitude of the primes has generally been regarded as elegant. It is a proof by contradiction, or, reductio ad absurdum, and it relies on an algorithm which will always bring in larger and larger primes, an infinite number of them. However, the proof is also subtle and has been misinterpreted by some with one well-known mathematician even remarking that the algorithm might not work for extremely large numbers. A long unsettled related problem, the twin primes conjecture, has also aroused the interest of many researchers. The author has been conducting research on the twin primes for a long time and had published a paper on them in an international mathematics journal in 2003. This informative paper presents some important facts on the twin primes which would be of interest to prime number researchers, with some remarks/reasons that point to the infinitude of the twin primes, including a reasoning which is somewhat similar to Euclid’s proof of the infinity of the primes; very importantly, 2 algorithms (refer to Appendix 3) for sieving out the twin primes from the
infinite list of the integers are also presented, which would be of interest to cryptographers and even computer programmers.
Category: Number Theory
[7] viXra:2107.0094 [pdf] replaced on 2021-07-28 03:58:32
Authors: A. A. Frempong
Comments: 5 Pages. Copyright © by A. A. Frempong
The author proves the original ABC conjecture which states that if A, B and C are three coprime positive integers such that A + B = C, and d is the product of the distinct prime factors of A, B and C, then d is usually not much smaller than C. The author will adhere to the wording of the original conjecture and not to any equivalent conjecture, since if one proves an equivalent conjecture, logically, one would also have to prove the equivalency, otherwise, the proof of the original conjecture would be incomplete. The author assumes that the statement " d is usually not much smaller than C" means the difference between C and d is usually less than a small positive number, say, ε.. Then, one obtains |C– d |< ε., which would be the conclusion. If A + B - C = 0, then for a very small positive number, δ > 0, one can write |A + B – C| < δ, From above, the hypothesis would be |A + B – C| < δ, and the conclusion would be |C– d |< ε. The author has proved that if |A + B – C| < δ, then |C– d |< ε.
Category: Number Theory
[6] viXra:2107.0078 [pdf] submitted on 2021-07-12 11:55:06
Authors: Zeolla Gabriel Martín
Comments: 10 Pages.
As there is no special primality test for Sophie Germain primes and safe primes as is the case with Fermat primes and Mersenne primes.
Argentest is born, a personal research project that develops a new exclusive deterministic primality test for Sophie Germain's prime numbers and safe prime numbers.
Category: Number Theory
[5] viXra:2107.0074 [pdf] submitted on 2021-07-13 02:01:25
Authors: Pranjal Jain
Comments: 12 Pages.
This paper generalizes problem 3 of the 2019 PROMYS exam, which asks to show that the last 10 digits (in base 10) of the n-th tetration of 3 are independent of ]n if n>10. The generalization shows that given any positive integers $a$ and b satisfying certain conditions, the last n digits (in base b) of the m-th tetration of a are independent of m if m>n.
We use numerical patterns as a guide towards the solution and explore an additional numerical pattern which shows a relation between decimal expansions and multiplicative inverses of powers of 3 modulo powers of 10.
Category: Number Theory
[4] viXra:2107.0072 [pdf] replaced on 2021-10-04 20:54:15
Authors: Denis Gallet
Comments: 7 Pages.
In this paper, I study particular values of Barnes G-function and we can simplify several integrals logarithm gamma.
Category: Number Theory
[3] viXra:2107.0059 [pdf] submitted on 2021-07-10 04:15:31
Authors: Alexey Ponomarenko
Comments: 7 Pages.
A lower bound is given for the number of primes in a special linear form less than N, under the assumption of the weakened Elliott-Halberstam conjecture.
Category: Number Theory
[2] viXra:2107.0044 [pdf] replaced on 2021-11-16 20:44:16
Authors: Gregory M. Sobko
Comments: 24 Pages. The title of the original submission has been changed.
A Recursive Algorithm described here generates consecutive sequences of Goldbach sets GP(m) = {p,p’| p + p = 2m} where p
and p’ are prime numbers and m >2, toward the proof of the Strong Goldbach Conjecture (SGC). The approach suggested here is based on the fundamental principle of mathematical induction and uses rather elementary set-theoretical technique and the property of shift-invariance of Goldbach sets with respect to specific mappings.
The main idea of this work is to develop a recursive algorithm for building the sequence of consecutive Goldbach sets {GP(k) | k = 3, 4, …, m} representing solutions to the system of Goldbach equations
{x + y = 2k| k is in [3, m]} in the intervals of integers I = [3, 2k – 3] The validity of the algorithm is based on the proved here recursive formula, generating Goldbach set GP(m) given the created consecutive sets GP(k), which are not empty for all
k = 3,4, …, m-1, due to inductive assumption.
The paper includes some discussion of the Diophantine geometry
of Goldbach sets related to Goldbach function, twin- and t-primes,
as well as the text of the computer script in R with realization of the suggested recursive algorithm.
Category: Number Theory
[1] viXra:2107.0033 [pdf] submitted on 2021-07-06 19:10:17
Authors: Babacar Gueye
Comments: 4 Pages. [Corrections are made by viXra Admin to comply with the rules of viXra.org]
This article talk about Brocard's problem. This problem consist to find the solution of the diophancian equation n! + 1 = m^2. Paul Erdos conjecture that the only solutions are the Brown numbers corresponding to the values 4, 5 and 7 of n. Using here on fondamental result of number theory and studing some fonction between whitch the neperian logarithm we proof that there are any solution corresponding to a value of n greater than 10.
Category: Number Theory
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