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[11] viXra:2212.0209 [pdf] replaced on 2026-03-03 03:10:29
Authors: Abdelmajid Ben Hadj Salem
Comments: 20 Pages. Comments welcome.
In this paper, we consider the $abc$ conjecture, we will give the proof that the conjecture $c$ smaller than $rad^{1.63}(abc)$ is true. It constitutes the key to resolve the $abc$ conjecture.
Category: Number Theory
[10] viXra:2212.0179 [pdf] submitted on 2022-12-24 01:36:58
Authors: Tae Beom Lee
Comments: 10 Pages.
The Riemann zeta function(RZF), ζ(s), is a function of a complex variable s = x + iy, which is analytic for x > 1. The Dirichlet Eta Function(DEF), η(s), is also a function of a complex variable s, which is analytic for x > 0. The zeros of RZF and DEF are all same. The Riemann hypothesis(RH) states that the non-trivial zeros of RZF is of the form s = 0.5 + iy. The clue of our proof stems from the symmetry properties of RZF zeros, stating that if there exists a zero whose real part is not 0.5, such as ζ(α+ iβ) = 0,0 <α < 0.5, also ζ(1-α+ iβ)=0, called the critical line symmetry. Then, the two symmetric zeros should be on the two edge lines of a strip α ≤ �� ≤ 1−α. In the strip there are infinitely many lines that are parallel to the edge lines. Our question was, when that strip is mapped by DEF, will these parallel relationships be kept? If the parallel relationships are kept, RH is true, if not, RH may be false. So, we identified four possible graphic patterns that may satisfy the critical line symmetry. We found that DEF can’t satisfy any of the four patterns. So, RH is true.
Category: Number Theory
[9] viXra:2212.0174 [pdf] submitted on 2022-12-23 02:14:34
Authors: Jan Norman Lawrence
Comments: 5 Pages.
A number should have a dimension. We can think of a real number as one dimensional number, we can think of a complex number as a two dimensional number. A dimension of a number should be n, where n is any positive integer of our choice. I will introduce algebra of n dimensional numbers for any n positive integer.
Category: Number Theory
[8] viXra:2212.0170 [pdf] submitted on 2022-12-22 10:49:23
Authors: Giovanni Di Savino
Comments: 4 Pages.
"A perfect number is a natural number which is equal to the sum of its divisors, also including the number one (but excluding the number itself)" and Euclid with an algorithm, (2^n -1)*2^(n-1 ) states that even perfect numbers are the result of the multiplication between two powers that both have the number 2 as a base and the indices of the powers differ by 1, i.e.: a power is 2^n -1 which is a prime number with the other power, 2^(n -1) which is an even number. The algorithm for even perfect numbers can be extended to odd perfect numbers which are the result of the multiplication between two powers that both have the same odd number as a base and the indices of the powers differ by 1, i.e.: a power is an odd number ^n -2 which is a prime number with the other power, odd number^(n -1) which is an odd number. Perfect even or odd numbers are the result of multiplying the result between two powers one of which is a prime number (obtained from a power). The difference between even and dispar perfect numbers is: a) for even perfect numbers the prime number is the result of a power of two minus 1; b) for odd perfect numbers the prime number is the result of a power of one of the infinite odd numbers minus 2.
Category: Number Theory
[7] viXra:2212.0162 [pdf] replaced on 2025-01-03 22:58:15
Authors: Theophilus Agama
Comments: 13 Pages. A few corrections have been implemented
Using the compression method, we prove an inequality related to the Gauss circle problem. Let $mathcal{N}_r$ denotes the number of integral points in a circle of radius $r>0$, then we have $$2r^2bigg(1+frac{1}{4}sum limits_{1leq kleq lfloor frac{log r}{log 2}floor}frac{1}{2^{2k-2}}bigg)+O(frac{r}{log r}) leq mathcal{N}_r leq 8r^{2}bigg(1+sum limits_{1leq kleq lfloor frac{log r}{log 2}floor}frac{1}{2^{2k-2}}bigg)+O(frac{r}{log r})$$ for all $r>1$. This implies that the error function $E(r)$ of the counting function $mathcal{N}_rll r^{1-epsilon}$ for any $epsilon>0.$
Category: Number Theory
[6] viXra:2212.0141 [pdf] submitted on 2022-12-18 03:01:52
Authors: Waldemar Puszkarz
Comments: 4 Pages. Originally posted on ResearchGate in December 2021.
Let 2p1p2 . . . pk−1 be an even squarefree Fibonacci number with k distinct prime factors. For each positive k, such numbers form an integersequence. We conjecture that each such sequence has only a finite number of terms. In particular, the factorization data for the first 1000 Fibonacci numbers suggests that there is only one such term for k = 2, 5 for k = 3, and 8 for k = 4. We also renew attention to the fact that a proof that there are infinitely many squarefree Fibonacci numbers remains lacking. Some approachto proving this, emerging from our study, is suggested.
Category: Number Theory
[5] viXra:2212.0125 [pdf] replaced on 2023-02-01 18:12:33
Authors: Jabari Zakiya
Comments: 12 Pages.
Since at least 1734 (when Euler solved the Basel problem), it’s been known for the positive even integers s, the Euler Zeta Function (EZF) can bewritten in terms of the even powers of Pi. I manipulate its form and find lurking (hidden) in it an exquisite and elegant formula for Pi . Thus, not only does the EZF have Pi embedded in it, Pi has embedded in its construction primorials of primes.
Category: Number Theory
[4] viXra:2212.0115 [pdf] submitted on 2022-12-11 02:58:52
Authors: Kwangsun Song
Comments: Pages.
Through solving the Collatz conjecture problem,I think about the problem of the modern mathematics. Without the modern mathematics,Through Pythagorean triangle How to find a large Pythagorean triangle, and Primitive Pythagorean triangle. Through Primitive Pythagorean triangle,How to find a large Pythagorean triangle, and Prime number. Through the interpretation of Fermat's Last Theorem, About what 'his surprising method of proof' is.
Category: Number Theory
[3] viXra:2212.0021 [pdf] submitted on 2022-12-04 01:29:06
Authors: Philippe Sainty
Comments: 14 Pages.
In this article, we define a " recursive local" algorithm in order to construct two reccurent numerical sequences of positive prime numbers (u2n) and (v2n), ((u2n) function of (v2n)), such that for any integer n≥ 2, their sum is 2n. To build these , we use a third sequence of prime numbers (w2n) defined for any integer n≥ 3 by : w2n = Sup(p∈IP : p ≤ 2n-3), where IP is the infinite set of positive prime numbers. The Goldbach conjecture has been verified for all even integers 2n between 4 and 4.1018. In the Table of Goldbach sequence terms given in paragraph § 10, we reach values of the order of 2n= 101000 Thus, thanks to this algorithm of "ascent and descent", we can validate the strong Euler-Goldbach conjecture.
Category: Number Theory
[2] viXra:2212.0011 [pdf] replaced on 2023-06-05 16:24:59
Authors: Theophilus Agama
Comments: 7 Pages. An updated version with more detailed explanation and proof.
Applying the pothole method on the factors of numbers of the form $2^n-1$, we prove the inequality $$iota(2^n-1)leq n-1+iota(n)$$ where $iota(n)$ denotes the length of the shortest addition chain producing $n$.
Category: Number Theory
[1] viXra:2212.0002 [pdf] submitted on 2022-12-01 01:34:22
Authors: Jean-Yves Boulay
Comments: 3 Pages.
A mathematical definition integrating, for any number, the notion of inferiority of divisor makes it possible to classify the number zero, the number one and all the primes in a unique set. Also, a complementary definition makes it possible to classify all the other whole numbers into a second unique set. Thus the set ℕ is considered to consist of two sets of whole numbers at absolute properties.
Category: Number Theory
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