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[14] viXra:2103.0199 [pdf] submitted on 2021-03-31 19:51:32
Authors: P.G. Bass
Comments: 15 Pages.
This paper provides a simple proof of Fermat’s Last Theorem via elementary algebraic analysis of a level that would have been extant in Fermat’s day, the mid seventeenth century. The proof is effected by transforming Fermat's equation to an nth order polynomial, which is solved for 5 cases revealing a pattern that enables an extrapolation to the general case.
Category: Number Theory
[13] viXra:2103.0198 [pdf] submitted on 2021-03-31 19:54:07
Authors: Juan Elias Millas Vera
Comments: 2 Pages.
In this article I wanted to contribute with some of my own investigations on the Collatz conjecture. Based on the ideas of parity and analyzing the conjecture from an algebraic point of view.
Category: Number Theory
[12] viXra:2103.0187 [pdf] submitted on 2021-03-30 20:25:34
Authors: Takamasa Noguchi
Comments: 7 Pages.
Explanation of how to find the quadratic residue.
Category: Number Theory
[11] viXra:2103.0181 [pdf] replaced on 2021-04-10 20:29:06
Authors: Olvin Dsouza
Comments: 9 Pages.
I have found an new method to factorize any certain large numbers p*q = n
products of numbers instantly.
This method works for any small or large product integers ‘n’ of any digits may
be millions, billions or even trillions of digits. This method is all about instantly
reversing (factorizing) and knowing the two factors i.e specially of two small
or large multiplied twin prime numbers or any terms (not divisible by 2 or 3)
and having a gap of two.
Category: Number Theory
[10] viXra:2103.0159 [pdf] replaced on 2023-12-20 19:42:51
Authors: Emmanuil Manousos
Comments: 23 Pages.
"Every natural number, with the exception of 0 and 1, can be written in a unique way as a linear combination of consecutive powers of 2, with the coefficients of the linear combination being -1 or +1" From this Theorem, four fundamental properties of odd numbers are implied: the conjugate of an odd number, the L/R symmetry, the transpose of an odd number and the octets of odd numbers. These concepts are used to obtain a classification of odd numbers and an algorithm for finding the factors of composite Fermat numbers.
Category: Number Theory
[9] viXra:2103.0158 [pdf] replaced on 2021-04-05 05:13:27
Authors: Carlos Castro, Ramon Carbo-Dorca
Comments: 17 Pages.
By establishing a dictionary between the QM harmonic oscillator and the Collatz process, it reveals very important clues as to why the Collatz conjecture most likely is true. The dictionary requires expanding any integer $ n $ into a binary basis (bits)
$ n = \sum a_{nl} 2^l $ ($l$ ranges from $ 0 $ to $ N - 1$) that allows to find the correspondence between every integer $ n $ and the state $ | \Psi_n \rangle $, obtained by a superposition of bit states $ | l \rangle $, and which are related to the energy eigenstates of the QM harmonic oscillator. In doing so, one can then construct the one-to-one correspondence between
the Collatz iterations of numbers $ n \rightarrow { n \over 2 }$ ($n$ even); $ n \rightarrow 3 n + 1$ ($n$ odd) and the operators $ {\bf L}_{ { n \over 2} }; { \bf L}_{ 3 n + 1 } $, which map $ \Psi_n $ to $ \Psi_{ { n \over 2 } }$, or to $ \Psi_{ 3 n + 1 } $, respectively, and which are constructed explicitly in terms of the creation $ {\bf a}^\dagger$, annihilation $ {\bf a }$, and unit operator $ { \bf 1 } $ of the QM harmonic oscillator. A rigorous analysis reveals that the Collatz conjecture is most likely true, if the composition of a chain of $ {\bf L}_{ { n \over 2} }; { \bf L}_{ 3 n + 1 } $ operators (written as $ L_*$ in condensed notation) leads to the null-eigenfunction conditions
$ ( {\bf L_* L_* \ldots L_* } - {\cal P } ) \Psi_n = 0 $, where $ {\cal P} $ is the operator that $projects$ any state $ \Psi_n $ into the ground state $ \Psi_1 \equiv | 0 \rangle $ representing the zero bit state $ | 0 \rangle$ (since $2^0 = 1$). In essence, one has a realization of the integer/state correspondence typical of QM such that the Collatz paths from $ n $ to $ 1$ are encoded in terms of quantum transitions among the states $ \Psi_n$, and leading effectively to an overall downward cascade to $ \Psi_1$. The QM oscillator approach explains naturally why the Collatz conjecture fails for negative integers because there are no states below the ground state.
Category: Number Theory
[8] viXra:2103.0150 [pdf] replaced on 2021-04-03 08:40:12
Authors: Theophilus Agama
Comments: 5 Pages. The constant taken as a function of $x$ has been removed, since the inequality is suggestive.
In this paper we obtain the estimate
\begin{align}
\# \left \{p\leq x~|~2p+1,p\in \mathbb{P}\right \}\geq (1+o(1))\frac{\mathcal{D}}{(2+2\log 2)}\frac{x}{\log^2x}\nonumber
\end{align}where $\mathbb{P}$ is the set of all prime numbers and $\mathcal{D}\geq 1$. This proves that there are infinitely many primes $p\in \mathbb{P}$ such that $2p+1\in \mathbb{P}$ is also prime.
Category: Number Theory
[7] viXra:2103.0136 [pdf] replaced on 2021-04-05 22:28:41
Authors: Mahdhi S. Si
Comments: 5 Pages. if you find this was interesting, and needs improvement, please left a message in comment section.
This paper proposed proof of Goldbach Conjecture by using a function such that the numbers occurences of conjecture solution in any even numbers can be estimated. The function sketches after Eratoshenes Sieve under modulo term such that the function fulfilled prime sub-condition in
closed intervals.
Category: Number Theory
[6] viXra:2103.0127 [pdf] submitted on 2021-03-18 21:02:47
Authors: Alexander Vasilievich Isaev
Comments: 16 Pages. [Corrections are made by viXra Admin to comply with the rules of viXra.org]
Монография от 24.03.2016 г., в которой автор приходит к выводу, что число Скьюза может быть порядка 8*10^60 (именно столько планковских времен помещается в возрасте Вселенной, которой 13,8 млрд лет).
A monograph dated 03.24.2016, in which the author comes to the conclusion that the Skuse number can be of the order of 8 * 10 ^ 60 (this is how many Planck times fit in the age of the Universe, which is 13.8 billion years old).[5] viXra:2103.0123 [pdf] replaced on 2021-03-25 15:48:10
Authors: Julian Beauchamp
Comments: 6 Pages.
The focus of this paper is the generalised Fermat equation, Pa^x + Qb^y=Rc^z, considered by Henri Darmon and Andrew Granville. It is closely related to a family of theorems and conjectures including the Fermat-Catalan Conjecture, the Darmon-Granville Theorem, the Beal Conjecture (also known as the Tijdeman-Zagier Conjecture) and Fermat's Last Theorem. We will consider these briefly before offering a proof that no solutions exist even for P,Q,R>1, for cases x,y,z>2, using a new binomial identity for a^x + b^y to an indeterminate power, z. The proof extends to its corollaries the Beal Conjecture and Fermat's Last Theorem.
Category: Number Theory
[4] viXra:2103.0076 [pdf] submitted on 2021-03-13 00:20:14
Authors: Alexander Vasilievich Isaev
Comments: 79 Pages. Книга (79 стр.), изданная автором весной 2003 г.. [Corrections are made by viXra Admin to comply with the rules of viXra.org]
В данной книге автором впервые употреблен термин "космология чисел" (позже он переродился в "числофизику"). Разумеется, что гениальный Леонард Эйлер стоит у истоков "космологии чисел" – это показано на примере удивительного мемуара Эйлера.
In this book, the author first used the term "cosmology of numbers" (later it was reborn into "number physics"). It goes without saying that the genius Leonard Euler stands at the origins of the "cosmology of numbers" - this is shown by the example of Euler's amazing memoir.[3] viXra:2103.0054 [pdf] submitted on 2021-03-10 19:45:15
Authors: Alexander Vasilievich Isaev
Comments: 47 Pages. [Corrections are made by viXra Admin to comply with the rules of viXra.org]
Метачисло, порожденное первыми простыми числами (2, 3, 5, 7, ..., Р, все они идут без пропусков),– это первое число в натуральном ряде, у которого первые делители являются КОПИЕЙ начала натурального ряда (1, 2, 3, 4, ..., Р, без единого пропуска). Впервые приведен алгоритм вычисления сколь угодно большого метачисла (нахождения его канонического разложения). По сути дела, это продолжение темы, начатой автором ещё в 2004 г. (в гл. 10 его «бумажной» книжки «Зеркало» Вселенной»).
The metnumber generated by the first prime numbers (2, 3, 5, 7, ..., P, they all go without gaps) is the first number in the natural series, in which the first divisors are a COPY of the beginning of the natural series (1, 2, 3, 4, ..., P, without a single gap). For the first time, an algorithm for calculating an arbitrarily large meta number (finding its canonical decomposition) is presented. In fact, this is a continuation of the theme started by the author back in 2004 (in Chapter 10 of his "paper" book "Mirror" of the Universe ").
Category: Number Theory
[2] viXra:2103.0044 [pdf] submitted on 2021-03-07 20:17:04
Authors: Zhang Tianshu
Comments: 13 Pages.
Regard positive integers which have a common prime factor as a kind, then the positive half line of the number axis consists of infinitely many recurring line segments which have same permutations of kinds of integer points, where ≥1. So let us make use of a positive half line of the number axis, but marked only with integer points at the positive half line.
After that, change gradually symbols of each kind of composite number points thereat according to the order of common prime factors from small to large, so as to form consecutive composite number points within limits that proven Bertrand's postulate restricts to prove Grimm’s conjecture.
Category: Number Theory
[1] viXra:2103.0038 [pdf] replaced on 2024-02-27 03:50:01
Authors: Kurmet Sultan
Comments: 9 Pages.
It is shown that Brocard’s Diophantine equation can have a solution only if the factorial is represented as a product of two natural numbers that differ by 2, or as a product of four consecutive natural numbers. Then a theorem was proven stating that the product of m consecutive natural numbers cannot be represented as a product of two natural numbers differing by 2 if m≠4. After this, it was proven that it is impossible to represent a factorial greater than 7! in the form of the product of four consecutive natural numbers and two natural numbers differing by 2, it follows that Brocard’s equation has no solutions, with the exception of the well-known three factorials.
Category: Number Theory
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