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| viXra.org > Number Theory > 2008 |
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[18] viXra:2008.0218 [pdf] submitted on 2020-08-30 10:33:45
Authors: Oliver Couto
Comments: 5 Pages.
Historically equation ( pa^n+qb^n=rc^n ) has been studied for degree 2 and eqn(pa^n+qb^n=rc^n ) here-in called equation (1) has been studied for n=2, p=1, q=9 (Ref.no. 4) by Ajai Choudhry. Tito Piezas has discussed about equation (1) when p=r=1 (Ref. no. 3). While Ref. no. (3 & 4) deals with equation no. (1) for degree n=2, this paper has provided parametric solutions for degree n=2,3,4,5 & 6. Also there are instances in this paper where parametric solutions have been arrived at using different methods.
Category: Number Theory
[17] viXra:2008.0206 [pdf] submitted on 2020-08-27 22:16:18
Authors: Zaizhao Meng
Comments: 3 Pages.
In this note, we generalize a lemma of Heath-Brown in the sieve method.
Category: Number Theory
[16] viXra:2008.0204 [pdf] submitted on 2020-08-28 09:20:26
Authors: Miguel Cerdá Bennassar
Comments: 4 Pages.
Este escrito es continuación del anterior de Julio de 2020. Las iteraciones de los números de las secuencias de Collatz, sustituyendo los números obtenidos por sus raíces digitales, se representan en el siguiente gráfico.
Category: Number Theory
[15] viXra:2008.0203 [pdf] submitted on 2020-08-28 09:25:19
Authors: Miguel Cerdá Bennassar
Comments: 1 Page.
Este gráfico representa los valores de las raíces digitales de los números pares en color azul y la de los números impares en color rojo de cualquier secuencia de Collatz. Las flechas indican el cambio
que experimentan los números en cada iteración.
Category: Number Theory
[14] viXra:2008.0202 [pdf] submitted on 2020-08-28 09:28:01
Authors: Miguel Cerdá Bennassar
Comments: 4 Pages.
Todas las secuencias formadas con la función de Collatz llegan siempre al número 16, y es después
de 4 iteraciones más, que llegan al número 1. ¿Porqué todas las secuencias llegan hasta el número
16? Este número puede resultar del número 32 o del número 5. Los dos son números de la forma 9n+5 y ambos tienen la misma raíz digital dr(5). El número 16 es de la forma 9n+7 y su raíz digital
es dr(7).
Category: Number Theory
[13] viXra:2008.0183 [pdf] submitted on 2020-08-24 05:05:49
Authors: Prateek P. Kulkarni
Comments: Pages.
The author proposes to derive a Formula that generates the sum of first n factorials
Category: Number Theory
[12] viXra:2008.0182 [pdf] submitted on 2020-08-24 05:31:47
Authors: Gerhard Frey, Abdelmajid Ben Hadj Salem
Comments: 14 Pages. In French.
It is a paper of Gerhard Frey published in 2012. It is an introduction about the $abc$ conjecture and its subtleties and consequences for the theory of numbers. It is a scientific version of the original paper.
Category: Number Theory
[11] viXra:2008.0154 [pdf] submitted on 2020-08-20 19:48:47
Authors: Alex-Pauline Poudade
Comments: 12 Pages. doi.org/10.7910/DVN/W8RRMA orcid.org/0000-0003-3037-1091
This paper discusses the mathematical Number Theory field of first order Prime-Generating Polynomials (P-GP) also referenced as Arithmetic progressions (AP). In regards to the Green-Tao theorem, it emphasizes “the primes contain arbitrarily long polynomial progressions” by exploring specifically the primorial p29# co-linear space.
Category: Number Theory
[10] viXra:2008.0149 [pdf] submitted on 2020-08-20 05:21:09
Authors: Yukihiro Sano
Comments: 16 Pages.
In the previous research paper “[2476] viXra:2008.0044 submitted 2020.8.7 Polynomials Generating Prime Numbers No.2”, I found many polynomials generating prime numbers, avoiding the value where the continuous prime number is interrupted in the prime numbers of vertical column of Euler's polynomial generating prime numbers to skip by 2 or 3 etc.. In this time, I also investigate polynomials generating prime numbers, avoiding the value where the continuous prime number is interrupted in the prime numbers of other famous polynomials generating prime numbers to skip by 2 or 3 etc..
As a result, I found many polynomials generating 29 to 10 consecutive prime numbers.
Category: Number Theory
[9] viXra:2008.0136 [pdf] submitted on 2020-08-19 10:54:45
Authors: T. Nakashima
Comments: 1 Page.
I show the 3rd possibility of Collatz problem.
Category: Number Theory
[8] viXra:2008.0115 [pdf] submitted on 2020-08-16 10:27:16
Authors: Seiji Tomita, Oliver Couto
Comments: 27 Pages.
Consider the below mentioned equation:
ax^4+ by^4+ cz^4+ dw^4 = 0-----(1)
In section (A) we consider solution’s with the condition on the coefficient’s of equation (1). Namely the product (abcd)=square. In section (B) we consider the coefficients of equation (1), with the product of coefficient’s (abcd) not equal to a square. Historically equation (1) has been studied by Ajai Choudhry, A. Bremner, M.Ulas (ref. 5) in 2014. Also Richmond (ref. 1 & 2) has done some ground work in 1944 & 1948. This paper has gone a step further, by finding many parametric solutions & new small numerical solutions by the use of unique Identities. The identities are unique, because they are of mixed powers(combination of quartic & quadratic variables) which are then converted to only degree four identities. As an added bonus in section (B), we came up with a few quartic (4-1-n ) numerical solutions for ( n < 50) by elliptical mean’s. A table of numerical solutions for the(4-1-n)equation arrived at by brute force computer search is also given (ref 7)
Category: Number Theory
[7] viXra:2008.0113 [pdf] submitted on 2020-08-15 20:49:40
Authors: Theophilus Agama
Comments: 6 Pages.
In this note we study the minimum overlap problem. We obtain the following crude inequality for the problem
\begin{align}
M(n)<\mathcal{D}(k)(1-o(1))\frac{n}{4}\nonumber
\end{align}where $\mathcal{D}(k)>1$.
Category: Number Theory
[6] viXra:2008.0054 [pdf] submitted on 2020-08-08 10:22:53
Authors: Aaron Chau
Comments: 4 Pages.
Looking for prime numbers has no relationship with zeta function all. For example, in ancient Greece in the West, Euclid proved that prime numbers are infinite and he used (multiplication and division) to express disproval; now in Eastern Hong Kong, this article also proves that the number of twin primes is infinite and the Riemann hypothesis is incorrect. The author uses (addition and subtraction to express more and less) to be eternal.
Category: Number Theory
[5] viXra:2008.0044 [pdf] replaced on 2020-09-10 01:30:58
Authors: Yukihiro Sano
Comments: 26 Pages.
In the Ulam spiral, there are places where prime numbers appear continuously on line. Integers are arranged in a square spiral in the Ulam spiral. I thought that if integers are arranged differently, other continuous prime numbers would appear. Therefore, I arrange integers in the angles of 45, 90, 135, 180, 225, 270, 315, 153, 160 degrees, hexagonal arrangement etc.. Then, prime numbers appeared continuously on line.
Looking at the prime numbers vertical column of Euler’s polynomial generating prime numbers, that I created in the hexagonal 90 degrees arrangement, I think how far the continuous numbers will continue if Euler prime numbers is selected to skip by 2 or skip by 3 etc., avoiding the value where the continuous prime number is interrupted, and I investigate. As a result, I found many polynomials generating 40 to 10 consecutive prime numbers.
Category: Number Theory
[4] viXra:2008.0043 [pdf] submitted on 2020-08-07 04:31:49
Authors: Harry K. Hahn
Comments: 1 Page.
A Fibonacci-Number-Sequences-Table was developed, which contains infinite Fibonacci-Sequences ( see explanatory Study ). The first number sequence F1 is the Fibonacci Main Sequence (1,1,2,3,5,8,13,..).The second number sequence in the table is the Lucas Sequence (1,3,4,7,11,18,..).
The developed ( natural ) Fibonacci-Sequence-Table shows interesting spatial dependencies between numbers of different Fibonacci-Sequences, which are connected to each other, by the golden ratio ( constant Phi ).
This table of Fibonacci Number Sequences can be extended towards infinity.
Note : Every natural numbers only appears once under the diagonal line in the table !
( in the lower left half of the table )
Category: Number Theory
[3] viXra:2008.0027 [pdf] replaced on 2020-08-11 19:31:51
Authors: Harry K. Hahn
Comments: 16 pages, 6 figures, 8 tables
A Fibonacci-Number-Sequences-Table was developed, which contains infinite Fibonacci-Sequences. This was achieved with the help of research results from an extensive botanical study. This study examined the phyllotactic patterns ( Fibonacci-Sequences ) which appear in the tree-species “Pinus mugo“ at different altitudes ( from 550m up to 2500m ) With the increase of altitude above around 2000m the phyllotactic patterns change considerably, the number of patterns ( different Fibonacci Sequences ) grows from 3 to 12, and the relative frequency of the main Fibonacci Sequence decreases from 88 % to 38 %. The appearance of more Fibonacci-Sequences in the plant clearly is linked to environmental ( physical ) factors changing with altitude. Especially changes in temperature- / radiation- conditions seem to be the main cause which defines which Fibonnacci-Patterns appear in which frequency.
The developed ( natural ) Fibonacci-Sequence-Table shows interesting spatial dependencies between numbers of different Fibonacci-Sequences, which are connected to each other, by the golden ratio ( constant Phi ) --> see Table
An interesting property of the numbers in the main Fibonacci-Sequence F1 seems to be, that these numbers contain all prime numbers as prime factors ! in all other Fibonacci-Sequences ≥ F2, which are not a multiple of Sequence F1, certain prime factors seem to be missing in the factorized Fibbonacci-Numbers ( e.g. in Sequences F2, F6 & F8 ).
With the help of another study ( Title: Phase spaces in Special Relativity : Towards eliminating gravitational singularities ) a way was found to express (calculate) all natural numbers and their square roots only by using constant Phi (ϕ) and 1. An algebraic term found by Mr Peter Danenhower, in his study, made this possible. With the formulas which I found, it seems to be possible to eliminate number systems and base mathematics only on Phi (ϕ) and 1 ( see my 12 conjectures )
Category: Number Theory
[2] viXra:2008.0004 [pdf] submitted on 2020-08-01 18:22:34
Authors: Derek Tucker
Comments: 3 Pages.
The Collatz conjecture is true, the 3n +1 problem generates a fractal spiral from odd multiples of three to elements 5 mod 8, if they are not already, and from there onto smaller elements already known to go to one. If T is the reduced Syracuse function, then if T(u) = x, so too does T(4u+1) = x. Also 3 mod 4 elements inevitabley map to 1 mod 4. All must descend.
Category: Number Theory
[1] viXra:2008.0003 [pdf] submitted on 2020-08-01 06:44:30
Authors: Yukihiro Sano
Comments: 25 Pages.
In the Ulam spiral,there are places where prime numbers appear continuously on line. Integers are arranged in a square spiral in the Ulam spiral. I thought that if integers are arranged differently, other continuous prime numbers would appear. Therefore, I arrange integers in the angles of 45, 90, 135, 180, 225, 270, 315, 153, 160 degrees,etc.. Then, prime numbers appeared continuously on line. I found many polynomials generating 19 to 6 consecutive prime numbers.
Category: Number Theory
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