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[18] viXra:2507.0190 [pdf] replaced on 2025-11-16 19:33:56
Authors: Marcin Barylski
Comments: 7 Pages. Updating results and the greatest prime island found so far (of size 16).
There are several interesting ways to depict distribution of primes like Ulam Spiral, Klauber Triangle or the Sacks Number Spiral. In all cases, Prime Number Theorem describes the asymptotic distribution of such numbers among the positive integers. This work is devoted to illustration of primes of form p x q +- C in a way that allows to search for clusters (so called islands of primes). The direct goal of this experimental work is to locate islands with the largest surface area and potentially discover some further patterns in distribution of primes.
Category: Number Theory
[17] viXra:2507.0186 [pdf] replaced on 2025-12-23 09:28:55
Authors: Bo Zhang
Comments: 10 Pages.
The Riemann hypothesis is proved true by finding two linear combinations of the real and imaginary parts of the Riemann $xi$-function.
Category: Number Theory
[16] viXra:2507.0155 [pdf] submitted on 2025-07-21 01:55:50
Authors: Theophilus Agama
Comments: 12 Pages.
We prove that a certain class of infinite sequences whose finite truncation is an addition chain must have a zero logarithmic density. This result is generic and can be applied to particular known infinite sequences with this property.
Category: Number Theory
[15] viXra:2507.0153 [pdf] submitted on 2025-07-21 21:18:33
Authors: Marko V. Jankovic
Comments: 6 Pages.
In this paper Riemann rearrangement theorem is going to be analyzed on a single example and it is going to be explained that the proof of the theorem is incomplete and wrong, That means that it does not matter how you rearrange the elements of the series, the sum would always stay the same. The reason that "rearranged" series does not have the same sum as the original series, is in the hidden omission of infinite number of elements that are contained in the original series. The content is presented in the form of explanation of a magic trick (since the claim of the theorem sounds as a real magic).
Category: Number Theory
[14] viXra:2507.0143 [pdf] replaced on 2025-12-15 02:13:35
Authors: Fawang Su
Comments: 15 Pages.
By setting the 3x + 1 iterative exponential orbit to construct and solve the initial number, and allowing the running orbit of this initial number to iterate under the control of the exponential orbit, this construction method proves that the step length of the 3x + 1 iterative orbit for specific types of numbers can be arbitrary or even infinitely long.
Category: Number Theory
[13] viXra:2507.0135 [pdf] replaced on 2025-07-25 21:35:53
Authors: Kuan Peng
Comments: 22 Pages.
Pythagorean triples are generated with Euclid’s formula. But how this formula was derived by or before Euclid is a mystery. We have derived Euclid’s formula directly from Pythagorean equation and classified all Pythagorean triples in a 3D table. The equation X^2+Y^j=Z^2 is proven to have infinitely many integer solutions. By comparing Pythagorean equation with Fermat’s equation for n=3 we were able to explain why Fermat’s equation with n=2 has integer solutions while with n 3 it has not. We propose an algebraic method to work Fermat’s last theorem.
Category: Number Theory
[12] viXra:2507.0134 [pdf] replaced on 2025-12-31 02:45:26
Authors: Izzie Boxen
Comments: 64 Pages.
The Sieve of Eratosthenes is taken as a definition of primes and is examined in a way that "opens" it into an array of rows labeled as primes and columns labeled as numbers. Through the introduced concept of prime candidates, numbers in each row that have the potential of being declared primes in lower rows, the opened Sieve reveals repeating and inter-related patterns of these prime candidates as well as other defined entities. These allow development of a number of relations that prove useful in examining the distribution of primes, with some new theorems being proved. Included is a new and independent proof for Bertrand’s postulate, two proofs for Lim((p_[n+1]-p_n)/p_n)=0, and proofs of conjectures by Brocard, Legendre, Andrica, and Oppermann.
Category: Number Theory
[11] viXra:2507.0116 [pdf] submitted on 2025-07-16 15:54:56
Authors: Theophilus Agama
Comments: 5 Pages.
In this paper, we formulate an optimization problem for addition chains.
Category: Number Theory
[10] viXra:2507.0110 [pdf] submitted on 2025-07-15 08:44:04
Authors: Simon Plouffe
Comments: 8 Pages.
A series of large-scale tests were performed on the first 1000 billion digits of the numberπ. First a direct visual test as well as a test using thousands of sequences from the OEIS catalog.The purpose of these tests is to detect possible patterns. Other tests were made on tenmathematical constants to 1 billion decimal places.
Category: Number Theory
[9] viXra:2507.0103 [pdf] submitted on 2025-07-14 11:05:05
Authors: Muhammad Razzaq Aman Wattoo, Mehar Ali Malik, Iqra Aftab
Comments: 6 Pages.
In this paper, we have developed a set S named as Aman’s set that is union of collection of sets generated by arithmetic sequence. Then we have used this set to generate two separate complete lists of prime numbers and composite odd numbers. Also we have run this set to find the finite list of the prime numbers and composite numbers in python successfully.
Category: Number Theory
[8] viXra:2507.0102 [pdf] submitted on 2025-07-14 20:41:24
Authors: Younghwan Yun
Comments: 11 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We propose a conservative structural framework to address the twin prime conjecture, aiming to demonstrate the unavoidable recurrence of twin prime pairs across the number line. By systematically applying the inclusion-exclusion principle within bounded intervals [p2 n−1, p2n), where pn is the n-th prime, we estimate the minimal lower bound of surviving 6k ± 1 pairs after sieving out all multiples of smaller primes. Our analysis shows that any composite survivor within such intervals would require a prime factor at least as large as pn, leading to a contradiction by exceeding the interval’s up-per bound. We derive an explicit minimal estimate Tn for the number of twin primepairs and show that it grows unboundedly with n. This non-probabilistic approach provides a concrete methodological pathway suggesting that the periodic sieve structure necessarily sustains infinitely many twin prime pairs, offering strong structural support for the twin prime conjecture.
Category: Number Theory
[7] viXra:2507.0092 [pdf] submitted on 2025-07-14 01:24:22
Authors: Hassan Bouamoud
Comments: 3 Pages.
This is a tentative [proof of] Andric's conjecture using a known result of Baker-Herman-Bintz.
Category: Number Theory
[6] viXra:2507.0066 [pdf] submitted on 2025-07-09 23:08:01
Authors: Theophilus Agama
Comments: 4 Pages.
Let ��u2099: su2080 = 1 < su2081 = 2 < ⋯ < su2095 = nbe an addition chain leading to n. Define the normalized profile vu2099(x) := s_{⌊xh⌋} / nfor any x ∈ [0,u202f1] and set xᵢ := i / h. We show that for any fixed x ∈ [0,u202f1] there exists an index i with 0 ≤ i ≤ h such that x·hu2044n ≤ vu2099(xᵢ) ≤ x + 1u2044h = x + o(1). This implies that no matter how an addition chain is built, at each fraction x ∈ [0,u202f1], there is some term whose normalized size is in the interval [x·hu2044n, x + o(1)]. This may be viewed as a Bertrand-type result in addition chains.
Category: Number Theory
[5] viXra:2507.0049 [pdf] replaced on 2025-07-17 20:32:38
Authors: Denis Micheal Odwar
Comments: 8 Pages.
For m ∈ Z, let N = 2m ≥ 8 and GN be a set of goldbach primes, p, of N defined as GN = {p : p ≤ N 2 andp ∤ N}. By denoting the cardinality of GN by |GN| or g(N), we show that ∀N, |GN| > 0, and the set of all these cardinalities, {|GN|}, is equal to the set of natural numbers i.e {|GN|} = N = {1,2,3,4,5,6,···}. We finally prove the famous binary |GN| Goldbach conjecture by showing that for all values of |GN|, i=1 µ(bi)Λ(bi) < 0, whenever bi = N −pi with pi ∈ GN,i ∈ N and 1 ≤ i ≤ |GN|. In particular we show that every N is a sum of two distinct primes.
Category: Number Theory
[4] viXra:2507.0029 [pdf] submitted on 2025-07-04 16:45:11
Authors: Walter A. Kehowski
Comments: 6 Pages.
The minimal set of primes in base b is a finite set of primes with the following property: if p is a prime, then there exists at least one element q of the minimal set such that the digits of q in base b form a subsequence of the digits of p in base b. The purpose of this note is construct the minimal set of primes in base 12.
Category: Number Theory
[3] viXra:2507.0028 [pdf] submitted on 2025-07-04 16:45:44
Authors: Theophilus Agama
Comments: 5 Pages.
We prove that the spacing between consecutive termsin an addition chain with non-decreasing τ -track can be generated by adding two previous terms in the chain.
Category: Number Theory
[2] viXra:2507.0011 [pdf] submitted on 2025-07-02 19:49:00
Authors: Bo Zhang
Comments: 10 Pages.
Goldbach's conjecture is proved through the new understanding of the fundamental theorem of algebra.
Category: Number Theory
[1] viXra:2507.0010 [pdf] submitted on 2025-07-02 23:44:41
Authors: Bahbouhi Bouchaib
Comments: 8 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This article presents a novel computational method for decomposing even numbers into sums of two prime numbers, a problem known as Goldbach’s Conjecture. Using optimized heuristics based on modular arithmetic and probabilistic constraints, I have developed a publicly accessible website capable of processing even numbers up to 1018 which is a historic record. My method, grounded in decades of theoretical insights and refined by computational efficiency, offers a new way to visualize and explore prime pair decomposition and contributes to the ongoing exploration of one of number theory’s most famous conjectures. The latest successful version of my new website for Goldbach's decomposition is available at: https://b43797.github.io/Bahbouhi-decomposing-Goldbach-conjecture2025/.
Category: Number Theory
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