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[13] viXra:2512.0128 [pdf] submitted on 2025-12-27 01:18:45
Authors: Arthur V. Shevenyonov
Comments: 4 Pages.
A simple-yet-fulfilling formula is presented showcasing a linkage between the prime number’s[shifted] index (ordinal rank) vs its signature trace, or power sum over the characteristic 2-basis. A loose analogy of minimum action or energy orbitals could be conceived as one way ofrationalizing the representation from amongst alternate, likewise 2-basis, candidates. A mostminimalist calculus is proposed accommodating such singular-basis travel.
Category: Number Theory
[12] viXra:2512.0111 [pdf] submitted on 2025-12-24 01:09:27
Authors: Theophilus Agama
Comments: 6 Pages.
We extend the inequality due to Alfred Brauer on standard addition chains to a sequence of additions leading to a finite number where at most at most $dgeq 2$ previous terms can be added to generate each term in the sequence.
Category: Number Theory
[11] viXra:2512.0110 [pdf] submitted on 2025-12-24 01:07:10
Authors: Arthur Shevenyonov
Comments: 3 Pages.
This paper proposes a ‘naïve,’ ‘three-line’ demonstration for RH building on a functional-equation reduction. It is suggested, inter alia, how prematurely restricting an exposition to the conjectured critical band a priori may usher in some paradoxic singularities that unduly restrict the complex candidate (imaginary extension) domain, albeit without questioning the real core. It remains to be judged whether the ‘gray area,’ as implied or straddled, qualifies as a ‘constructive’ completion of the RH around its frontier of inference.
Category: Number Theory
[10] viXra:2512.0109 [pdf] submitted on 2025-12-24 01:08:41
Authors: Arthur Shevenyonov
Comments: 7 Pages. (Note by viXra Admin: Please cite and list scientific references)
A ‘naïve’ look into simple & augmented natural power-difference forms (nPDF) unleashes implications & patterns in areas as diverse as, the RH, primality formulae, and structure parallels, to name but a few.
Category: Number Theory
[9] viXra:2512.0090 [pdf] submitted on 2025-12-19 21:58:53
Authors: Farhad Aliabdali
Comments: 7 Pages. (Note by viXra Admin: Please cite listed scientific references and submit article written with AI assistance to ai.viXra.org)
This paper presents a complete, closed-form mathematical equation that exactly computes the prime-counting function π(N) for any integer N ≥ 2. Unlike existing methods which are either asymptotic approximations or recursive algorithms, our formulation is a single evaluable expression. The equation operates in two distinct modes: (1) using a sequence of known primes, or (2) using the simple sequence Ju2081 = 2, Ju2099 = (n-1)-th odd integer ≥ 3 for n ≥ 2, with an intrinsic primality test μu2099 = ⌈∏u2093u208cu2081u207fu207b¹ (Ju2099/Ju2093 - ⌊Ju2099/Ju2093⌋)⌉ where μu2099 = 1 if and only if Ju2099 is prime. The formula directly yields π(N) through elementary arithmetic operations without recursion, iteration, or algorithmic procedures. The implications of this formula are explored in comparison to existing prime counting functions and its potential impact on the study of prime distribution, it is an explicit sieve-theoretic expression and a self-contained rewriting. This complements classical exact prime-counting methods (Meissel—Lehmer and descendants), which are vastly more efficient for computation.
Category: Number Theory
[8] viXra:2512.0089 [pdf] replaced on 2026-01-28 06:30:48
Authors: Umberto Bartocci, Alessandro Miotto
Comments: 5 Pages.
Given any natural number n, we consider the subset C(n) of all natural numbers which could be written in decimal basis as a string of length n (n-numbers). We are looking for subsets of C(n) which appear interesting in connection with n-prime numbers and twin n-prime numbers too. Those numbers consisting of only the digits 1 and 7 (that we call Sofia numbers) appear to be quite promising in this context, and their study suggests some natural conjecture.
Category: Number Theory
[7] viXra:2512.0079 [pdf] submitted on 2025-12-17 06:26:11
Authors: Minho Baek
Comments: 2 Pages.
The purpose of this paper is to introduce regularity on Mersenne number with n=odd number. There is regularity among n=odd numbers in Mersen number. If the k and l is odd number, 2^kl-1=α(2^k-1).
Category: Number Theory
[6] viXra:2512.0070 [pdf] submitted on 2025-12-16 03:29:37
Authors: Farhad Aliabdali
Comments: 13 Pages. (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)
The Collatz map T(n)=n/2 for even n and T(n)=3n+1 for odd n admits classical affine descriptions via parity vectors, but these typically compress each odd event into the macro-step (3n+1)/2, obscuring intermediate algebraic states. We introduce a two-stage expansion that separates an odd event into a rewrite step R (expressing n=2x+1) followed by a forced follow-up C (sending x↦3x+2), alongside the even halving step E. This yields a word system over {E,R,C} and a uniform normal formX_N (w)=(3^k(w) X_0+2^D(w) -3^k(w) +σ_N (w))/2^D(w) ,where σ_N (w) admits an explicit signed monomial expansion in powers of 3 and 2. We prove that complete two-stage words (those with every R immediately followed by C) compress under RC↦O to the standard parity-vector affine form, giving a precise equivalence criterion and a canonical matching rule (k,D,Σ). Consequently, removing the standard-image equations from the two-stage enumeration leaves exactly the truncated (dangling-R) equations corresponding to intermediate states not representable in the standard form. Finally, we derive residue-class "locking" conditions modulo 2^D(w) , clarifying integrality constraints and connecting the framework naturally to 2-adic formulations.
Category: Number Theory
[5] viXra:2512.0065 [pdf] submitted on 2025-12-16 02:33:57
Authors: Zhi Li, Hua Li
Comments: 8 Pages.
Hyperreal numbers are a field encompassing real numbers, infinitesimals, and infinities. In thehyperreal number system, infinitesimals are not equal to zero, and operations can be performedon infinitesimals and infinities.Based on hyperreal number theory, this paper discovers that multi-level radicals that cannot besimplified have no definite value, varying in size; these can be called quantum numbers or inaccurate numbers. A special type of multi-level radical can exhibit quantum superposition phenomena similar to those in the physical world. This allows the hyperreal number system to be extended to a supernumber system, a field encompassing real numbers, infinitesimals, infinities,and inaccurate numbers.Confirming the existence and properties of quantum numbers or inaccurate numbers and classifying them as supernumbers provides a new perspective. Within this framework, the negation proves the Riemann hypothesis.
Category: Number Theory
[4] viXra:2512.0054 [pdf] submitted on 2025-12-12 00:41:14
Authors: Laurent Nedelec
Comments: 25 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This new text on the Collatz/Syracuse problem is a continuation of the document published in February 2025 on viXra (Laurent Nedelec : An algorithmic approach to solving the Collatz/Syracuse problem. viXra 2502.0056). It explains why the probabilities of divergence for Syracuse trajectories are extremely small. The same conclusion is obtained for non-trivial cycles: their existence is nearly impossible. These two results reinforce the conclusion of the previous text—reached through different methods—namely that the Collatz conjecture is, with very high probability, true.In this new work, we first analyze the structure of the alternations between even and odd iterations within Syracuse trajectories. We then show that N* has an equiprobable structure with respect to even and odd iterations in Syracuse trajectories. Next, we examine how this equiprobability within the structure of N* leads trajectories to be globally decreasing. Finally, we address the implications of these results for the questions of divergent trajectories and non-trivial cycles.
Category: Number Theory
[3] viXra:2512.0052 [pdf] submitted on 2025-12-10 09:40:35
Authors: Miroslav Sukenik, Magdalena Sukenikova
Comments: 4 Pages.
The calculation of this constant requires advanced numerical methods such as interval mathematics, using specialized software for high accuracy and tight error limits. In this article, we present a simpler method for deriving the Viswanath’s constant.
Category: Number Theory
[2] viXra:2512.0039 [pdf] submitted on 2025-12-08 05:43:00
Authors: Minho Baek
Comments: 4 Pages.
The purpose of this paper is to introduce patterns in prime numbers and then primality test from those patterns. Those patterns are the regularity among odd numbers for eliminating the composite number such as Sieve of Eratosthenes. Those patterns shows the regularity among the odd numbers except 2. Those patterns may be used to determine which odd numbers are prime numbers. This pattern may reduce the computation time to find prime numbers in some number range.
Category: Number Theory
[1] viXra:2512.0030 [pdf] submitted on 2025-12-07 20:22:13
Authors: Ansh Mathur
Comments: 8 Pages. (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)
This paper introduces the Universal Divisibility Framework (UDF), a comprehensive mathematical theory that extends the classical notion of divisibility from integers to rationals, reals, and complex numbers. The framework is built upon the Universal Divisibility Function $d(a, b, c) = lfloor a/c floor (b bmod c) - lfloor b/c floor (a bmod c)$, which provides a unified criterion for divisibility across multiple number systems. We establish the Universal Divisibility Theorem, proving that for $a, b in mathbb{R}$ with $b eq 0$, and an integer $c$ satisfying $lfloor b/c floor = pm 1$, we have $b mid a$ if and only if $d(a, b, c) equiv 0 pmod{b}$. This framework not only recovers all classical integer divisibility rules as special cases but also eliminates false positives that arise when traditional rules are naively extended to non-integer domains. We provide explicit divisibility formulas for numbers 1—1000, demonstrate applications to Diophantine equations and matrix algebras, and discuss implications for computational number theory and cryptography.
Category: Number Theory
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