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[13] viXra:2607.0129 [pdf] submitted on 2026-07-31 20:39:24
Authors: Taha Muhammad
Comments: 13 Pages.
A Collatz sequence is a series of numbers generated by taking any positive integer and repeatedly applying two simple arithmetic rules [Collatz Sequence Rule (CSR)] until the number reaches 1:1- If the current number is even, divide it by 2: (n /2). 2- If the current number is odd, multiply it by 3 and add 1: (3 n + 1). Taha^' s Collatz Sequence Fact 1 (TCSF1): if any sequence S(n)={a,b,c,u2026x,u2026,t}u2026by sequence rule ⇒lS(n)=lS(a)=lS(b)=⋯lS(x)=lS(t)u2026by (TCSF1) This paper presents a systematic evaluation of the localized boundary layers governing the classical Collatz Conjecture ((3n+1)) utilizing Taha’s Sketch and a comprehensive Loop Table state matrix. We evaluate the algebraic behavior of the sequence within an isolated loop structure containing exactly (r) elements, where (k) is defined strictly as a positive integer serving as the cofactor of 3 when (r = 3k) ((r, k in mathbb{N}^+)). By partitioning the entire positive integer domain into two mutually exclusive sets ((mathbb{N}_{3k} cup mathbb{N}_{text{non}_3k} = mathbb{N}^+)), we solve the general loop equation (x = text{(potential loop element)}) to show that any alternative configurations outside of this strict ratio consistently yield fractional values, breaking integer closure properties ((x otin mathbb{N}^+)). Consequently, we demonstrate that alternative or non-trivial integer loops are structurally impossible, concluding that all trajectories over the valid domain must uniquely collapse to the classical ({1, 4, 2}) attractor cycle.
Category: Number Theory
[12] viXra:2607.0120 [pdf] submitted on 2026-07-29 14:47:14
Authors: George Athanasiou
Comments: 9 Pages.
This paper explores the normalization of Fourier series kernelsassociated with elementary arithmetic functions. By analyzing keyfunctions such as divisor related functions, the prime gap function andthe inverse prime counting function. With these results, a framework fortheir spectral decomposition in terms of complex exponential functionswas developed. Through contour integration techniques and normalizationof the Dirac delta function, new methods can be derived from the analyticstructure of these functions. These findings contribute to the broaderunderstanding of arithmetic functions and their link to Fourier series.
Category: Number Theory
[11] viXra:2607.0105 [pdf] submitted on 2026-07-26 02:01:26
Authors: Francesco Aquilante
Comments: 15 Pages.
We present a complete, unconditional proof of Beal's Conjecture by introducing a novel hybrid framework that bridges classical Euclidean geometry, the rational parametrization of orthogonal vectors, and Gaussian integer arithmetic. By mapping a hypothetical coprime counterexample triple $a^k + b^m = c^n$ onto a two-dimensional geometric construction split by a rational altitude, we derive two fundamental algebraic identities. We show that the geometric sign clash inherent to these identities can be non-trivially satisfied only within the Gaussian integer domain $mathbb{Z}[i]$. Applying localized $pi$-adic valuation analysis reveals a powerful, invariant parameter symmetry relation across the coordinate space. Systematically evaluating this symmetry across inert ($3 pmod 4$), split ($1 pmod 4$), and ramified ($q=2$) prime instances demonstrates that any valid parameter distribution forces a scale explosion ($a^k > b^{2m}$).This parameter growth contradicts the strict mathematical limits required to bridge consecutive perfect powers, proving that no such counterexamples can exist and completing the proof.
Category: Number Theory
[10] viXra:2607.0101 [pdf] submitted on 2026-07-23 12:12:44
Authors: Payam Danesh
Comments: 16 Pages.
Ramanujan primes encode a persistent form of prime abundance: after the nth threshold, every interval of the form (x/2,x] contains at least n primes. Existing work establishes their existence, global bounds, and leading asymptotic equivalence to the 2nth prime, yet the exact discrete mechanism and the next asymptotic correction are usually treated separately. This work provides a general framework for the analysis of stability and thresholds. It uses a prime window counting process as an event-driven step function and describes the latest deficit as well as the finite sieve process with a known upper bound guaranteeing its completeness. Asymptotic inversion of the prime-counting expansion formula provides the second-order approximation of the threshold with respect to a constant window value c from the interval (0, 1). In the classical case, the approximation shows that the difference between the 2n-th prime number and 2n converges to log 2. Exact computation of the first 100,000 Ramanujan primes confirms the predicted scale; at n=100,000, the second-order approximation has relative error 3.37×10^4. The framework separates rigorous asymptotics, certified finite computation, and historical terminology, providing a reproducible basis for further explicit estimates.
Category: Number Theory
[9] viXra:2607.0094 [pdf] submitted on 2026-07-25 02:18:45
Authors: Jatin Dharkar
Comments: 4 Pages. (Note by viXra Admin: Please cite and listed scientific references)
This paper introduces an algebraic sieve for identifying Gaussian primes within arithmetic progressions of the form N = ax + b. By mapping the primality of N onto a one-dimensional integer index x, we establish a parameterization x = (ap + b)n + p. We extend the domain of the parameters p and n to the field of Gaussian rationals Q(i), subject to specific integrality constraints on the resulting factors. We provide a formal proof of the sieve's completeness, demonstrating that the existence of a valid parameter pair (p, n) is both a necessary and sufficient condition for the compositeness of ax + b within the ring of Gaussian integers Z[i].
Category: Number Theory
[8] viXra:2607.0089 [pdf] submitted on 2026-07-22 03:04:24
Authors: Walter A. Kehowski
Comments: 9 Pages.
The author observed belatedly that {9,4} is the spectral basis of n=sigma(9)−1=12, where sigmau2032(9)=sigma(9)−9=4, the aliquot sum of 9. The author provides examples of m such that {m,sigmau2032(m)} is the spectral basis of n=sigma(m)−1, called the aliquot spectral basis of n.
Category: Number Theory
[7] viXra:2607.0079 [pdf] submitted on 2026-07-20 19:52:26
Authors: Joerg Moesch
Comments: 70 Pages.
This paper is about the arrangement and stacking of equal-sized spheres into geometric figures, and the sequences an remarkable yet largely unknown insights that can be obtained as a result.
Category: Number Theory
[6] viXra:2607.0070 [pdf] submitted on 2026-07-17 06:51:51
Authors: A. A. Frempong
Comments: 18 Pages. Copyright © by A. A. Frempong
Collatz conjecture states that beginning with a positive integer, if one repeatedly performs the following operations to form a sequence of integers, the sequence will eventually reach the integer one; the operations being that if the integer is even, divide it by 2, but if the integer is odd, multiply it by 3 and add one; and also, use the result of each step as the input for the next step One would note the patterns of the sequence terms as the Collatz process reaches the equivalent powers, 2^(2k) (k = 2, 3,...) and the sequence reaches the integer 1 by repeated division by 2. Two main cases are covered. In Case 1, the integer can be written as a power of 2 as 2^(k) (k=1,2,3,u2026), and in this case, the sequence would reach the integer one by repeated division by 2, i.e., 2^(k-1), 2^(k-2), 2^(k-3,),u2026,2^(k-k). In Case 2, the integer cannot be written as a power of 2, but the sequence terms reach the equivalent power, 2^(2k) (k = 2, 3,...) and by repeated division by 2, the sequence will reach the integer 1. In Case 2, when the sequence terms reach some particular integers such as 5, 21 and 85, the application of 3n + 1 to these integers will result in the powers, 2^(2k). One would call these integers, the 2k-power converters. There are infinitely many 2k-power converters as there are 2^(2k) powers. There are infinitely many paths for converting integers to 2^(2k) powers. Of these paths, the integer 5-path, is the nearest 2^(2k) converter path to the integer 1 on the 2^(2k)-route. Other integers can follow the integer 5-path to 16 as follows: Let n be an integer whose sequence terms would reach 16, and let n ± r = 5, where r is the net change in the sequence terms before the integer 5; and one uses the positive sign if n<5, but the negative sign if n > 5. One will call the following, the 5-path 2k-converter formula: 3(n ± r) + 1 = 16. By the substitution axiom, using this formula, the sequence of every positive integer that cannot be written as a power of 2, would reach the integer, 16, and continue to reach the integer 1. Therefore, the sequence of every positive integer would reach the integer 1.
Category: Number Theory
[5] viXra:2607.0061 [pdf] submitted on 2026-07-15 21:57:59
Authors: Payam Danesh, Raoul Bianchetti
Comments: 6 Pages.
We present a rigorous formulation of that idea through shifts generated by the continuous Gram function. We study the set of parameters τ for which ζ( s+itτ) uniformly approximates ζ( s) on compact subsets of D with connected complements. The method is based on Gram-shift universality, weak convergence of probability measures on the space of analytic functions and the support theorem for the random Euler product. The support is precisely the analytic functions that vanish nowhere in D and the identically zerofunction. This structure leads to two precise equivalences: The Riemann hypothesis is true if and only if the Gram-shift self-approximation holds with positive lower density for each admissible compact set and each approximation radius; and equivalently, when the limiting density exists and it is positive except at mostcountably many radii.
Category: Number Theory
[4] viXra:2607.0060 [pdf] submitted on 2026-07-13 07:59:26
Authors: Rolando Zucchini
Comments: 13 Pages.
This paper reports the most significant studies on the knowledge of Pi-Greek. The number that has stimulated mathematicians throughout the centuries, from Archimedes to Descartes to Euler. It was the numerous attempts to prove the squaring of the circle that led to increasingly precise approximations of Pi. The proofs continued even after the Royal Academy of Sciences in Paris, in 1775, abandoned examining the many alleged "solutions" to this problem. The irrationality of π was demonstrated by Ferdinand von Lindemann in 1882. This article is mainly a historical research on the number Pi-Greek.
Category: Number Theory
[3] viXra:2607.0034 [pdf] submitted on 2026-07-11 23:16:01
Authors: Walter A. Kehowski
Comments: 8 Pages.
Let tau(n) be the number of divisors of n and let phi(n) denote the number of relatively prime numbers less than n. A number is called refactorable if and only if tau(n) divides n. It is known that phi(n) divides n if and only if n is 3-smooth. Furthermore, n/phi(n)=2 when n is a power of two and n/phi(n)=3 otherwise. Define lambda(n)=lcm(tau(n),phi(n)). Then n/lambda(n) is in {1,2,3} and n is called a tauphi-number. It is the purpose of this note to show when lambda(n)=n, called the unital tauphi-numbers.
Category: Number Theory
[2] viXra:2607.0010 [pdf] submitted on 2026-07-03 10:28:12
Authors: Payam Danesh
Comments: 9 Pages.
We studied the Robin defect associated with the inequality σ(n)<e^γ nlogu2061logu2061n, express its Laplace transform through Ramanujan’s transformation for the divisor Lambert series and isolate the precise difference between smoothed positivity and coefficientwise positivity. The main results give us an exact Ramanujan-transformed identity for the Robin defect, an equivalent coefficientwise formulation of the Riemann Hypothesis showing why transform-level positivity cannot by itself prove the hypothesis and an extremal reduction to colossally abundant and highest abundant numbers. Numerical data for early extremal integers illustrate how the normalized defect behaves past the exceptional value 5040. Ramanujan’s identities provide powerful global control, but the Riemann Hypothesis requires pointwise positivity at the extremal divisor-rich integers.
Category: Number Theory
[1] viXra:2607.0007 [pdf] submitted on 2026-07-04 02:50:07
Authors: Christoper Mututu
Comments: 36 Pages.
We study a map T on the integers defined by T(n)=n^2+1 when n is prime or even and T(n)=n/P(n) when n is odd composite, where P(n) denotes a designated prime factor n. Two variants arise according to the choice of P(n). In Part I, P(n) is the largest prime factor of n while in Part II, it is the smallest. Informally, the conjecture of this paper asserts that every integer greater than one in absolute value eventually enters the single twelve element cycle5→26→677→458330→210066388901→52357→41→1682→2829125→1625→125→25 and remains there forever. Formally, we verify this for every integer n∈[2,1000] under Part I with no exception and no alternate cycle observed and we conjecture it holds for every integer n in the domain Z^*=Z {-1,0,1} but we do not prove it.Toward this conjecture, we prove two theorems. The first is exact rather than asymptotic. For any odd composite m, repeated application of the largest prime factor reduction reaches a prime in precisely Ω(m)-1 steps where Ω(m) counts the prime factors of m with multiplicity. Each application removes exactly one element from the prime factorization multiset m so the count decreases by exactly one per step and terminates uniquely at a prime. The second theorem follows from the first. Under the extension of primality to negative integers through |n|, which is the only convention under which the conjecture is well posed on Z, every negative integer reaches a positive value within at most Ω(|n|) steps. This reduces the negative integer case of the conjecture entirely to the positive integer case. We also identify an obstruction to further verification that is structural rather than a matter of computing resources. The reduction step of T requires the complete factorization of the input which is a problem for which no general sub exponential algorithm is known. Repeated application of the squaring branch can therefore produce integers whose factorization lies beyond any presently known method regardless of computing time available. Our verification required factoring intermediate values of up to 96 digits and succeeded in every instance though with no guarantee that a harder instance does not arise beyond the tested range. For Part II, this obstruction is severe enough to foreclose even a conjecture. Every trajectory examined exceeded 100 digits within fewer than 25 iterations without any value repeating. We are unable to characterize the long-term behavior of Part II by any method available to us and as a result, Part II is entirely open.
Category: Number Theory
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