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[9] ai.viXra.org:2603.0095 [pdf] submitted on 2026-03-24 23:33:00
Authors: Mohammad Aldebbeh
Comments: 5 Pages. 2 figures (Note by viXra Admin: Please cite and listed scientific references)
Legendre's Conjecture states that there is a prime number between n2 and (n+1)2 for every positive integer n. This paper proposes a symmetrical, computationally verified strengthening of Legendre's Conjecture by anchoring the search for primes exclusively on even squares. We hypothesize that for every even integer n, there exists a prime q < 2n+1 such that n2+q is prime, and a prime q < 2n-1 such that n2-q is prime. We demonstrate that these two conjectures collectively imply Legendre's Conjecture for all integers. Furthermore, we explore the algebraic links to Polignac's and Goldbach's Conjectures, highlighting how our tight bounding of q bridges these theories. Probabilistic heuristics based on the Prime Number Theorem are provided to justify the logarithmic growth of q. Finally, we present computational evidence verifying both conjectures up to n=108.
Category: Number Theory
[8] ai.viXra.org:2603.0089 [pdf] submitted on 2026-03-22 12:27:12
Authors: Ruben Gafencu
Comments: 14 Pages.
We present a self-contained combinatorial and sieve-theoretic framework forestimating and bounding the number of primes of the form $x^2+1$ in a naturalfamily of intervals. The centrepiece is the emph{Coprime Identity}: for anyinteger $m$ with $n < m le n^2$, $m$ is prime if and only if $m$ is coprimeto the primorial $P_n = prod_{p le n} p$. Applying this identity to thepolynomial $f(x) = x^2+1$ and encoding the root structure of $f(x)bmod p$via quadratic residue theory yields the emph{Quadratic Totient Estimate}[ mathcal{N}(n) ;=; frac{n}{2} prod_{substack{ple n pequiv 1pmod{4}}} !frac{p-2}{p},]which is strictly increasing and diverges to infinity. We then introduce theemph{Quadratic Jacobsthal function} $gQ(n)$, the maximum gap betweenconsecutive survivors of the quadratic sieve for $f$, and give a sieve-theoreticargument --- based on the computation of the sieve dimension $kappa = 1$ andthe linear sieve lower bound --- that $gQ(n) < n^2 - n$ for all sufficientlylarge $n$. Large-scale numerical verification confirms the formula's accuracy:against exact prime counts from OEIS~A002496, the relative error of$mathcal{N}(n)$ remains below $8%$ for $n$ up to $10^{13}$, with systematicimprovement toward zero as a tail-factor correction is applied. We discuss theconnection to the Hardy--Littlewood Conjecture~F, the Bateman--Horn singularseries, and Iwaniec's 1978 $P_2$ theorem, and we carefully delineate theremaining obstacles --- the parity barrier and the rigorous control of theerror term --- that separate this heuristic from a complete proof of Landau'sFourth Problem.
Category: Number Theory
[7] ai.viXra.org:2603.0080 [pdf] submitted on 2026-03-18 13:03:06
Authors: Wiroj Homsup
Comments: 3 Pages.
A modification of the Sundaram sieve of twin primes is introduced. A twin prime pair is obtained from the set D which consists of elements n such that n not representable as the forms 2ij + i + j and 2ij + i + j -1 for positive integers i, j. An element n is in D if and only if (2n+1, 2n+3) is a twin prime pair. This sieve algorithm can find the number of twin primes below a certain value.
Category: Number Theory
[6] ai.viXra.org:2603.0077 [pdf] submitted on 2026-03-17 23:52:15
Authors: Khazri Bouzidi Fethi
Comments: 5 Pages.
We consider a matrix constructed from a finite set of prime numbers and a real parameter. The largest eigenvalue of this matrix is expressed as half the sum of two quantities: a sum independent of the parameter and the modulus of an exponential sum over primes. We prove two unconditional results:· The logarithm of this eigenvalue is a convex function of the parameter.· A certain symmetric product, formed from this eigenvalue and its value at the symmetric point, attains its global minimum at a remarkable point—the critical half-line.The proofs rest on an exact algebraic identity and the Cauchy-Schwarz inequality. No hypothesis on the zeros of the zeta function is used.This work introduces an original spectral object associated with a finite set of prime numbers. The angular Gram matrix, of rank at most two, encodes correlations between prime powers through oscillating phases. Its variational study reveals convexity and symmetry properties that naturally distinguish the critical line sigma = 1/2. These results, entirely unconditional, offer a new framework for exploring connections between spectral structures and the distribution of prime numbers. They also open perspectives on asymptotic behavior as the set size tends to infinity—a question that remains open.
Category: Number Theory
[5] ai.viXra.org:2603.0066 [pdf] submitted on 2026-03-15 13:37:31
Authors: Jungwon Park
Comments: 53 Pages.
We present a computer-assisted proof that every positive integer orbit under the Collatz map $T(n) = n/2$ (if $n$ is even), $(3n + 1)/2$ (if $n$ is odd) eventually reaches 1. The argument consists of two independent parts, H1 and H2, each of which reduces an originally infinite verification problem to a finite exact certification step. The mathematical reductions are traditional proofs; the certifications are finite, deterministic computations using only exact integer and rational arithmetic, whose complete specifications (input, algorithm, output) are given in the appendices.For H1 (divergent-orbit exclusion), a deterministic normalized block-drift analysis establishes an exact affine envelope that rules out all parameter regimes below an explicit threshold $V_{env} approx 1.5990$. The remaining oscillatory regime is reduced to a finite certification problem over a quotient-state automaton whose mathematical soundness is guaranteed by a chain of five theorems (Theorems 12.8—12.14). The certification is discharged by exhaustive exact computation (Theorem 12.17).For H2 (non-trivial cycle exclusion), a gate-based reformulation converts the cycle problem into an explicit inequality system. Discrete convexity arguments and a two-value support reduction compress the problem to a single-parameter bound, which is then verified by exact rational arithmetic over a rigorously delimited feasible range. The computational components are finite, exact, and logically isolated. Their role is analogous to the finite case verifications in other computer-assisted proofs such as the Four Colour Theorem and the Kepler Conjecture: the mathematical argument provides a complete reduction to finite exact certification problems, and the computations discharge those problems. The complete source code for both certifications, together with their full execution logs and exact rational outputs, is included in the appendices (Appendices E—G).
Category: Number Theory
[4] ai.viXra.org:2603.0059 [pdf] submitted on 2026-03-13 02:27:42
Authors: Rajat Yadav
Comments: 14 Pages.
The note separates three layers: (i) current literature status, (ii) rigorous reductions andobstruction theorems derived in the thread, and (iii) conditional routes and dead ends. Thecentral conclusion is negative but sharp: many natural corridors — classical Apéry clearing,the EMN simple-pole and polynomial higher-pole programs, fixed-base q-difference compression,several modular/holonomic packetizations, and a broad discrete moment/Hankel family— fail for precise structural reasons. The revised version also records five structural ingredientsthat were missing from the earlier draft: a four-jet modular rigidity theorem for the X1(5)Picard-Fuchs engine, a noncharacteristic edge-splitting theorem for higher poles, injectivity/isomorphism theorems for the EMN collapse operators, a boundary-annihilator theoremfor the edge packet, and formal torsion-support / multiplicative-complexity obstructions onthe modular q-side.
Category: Number Theory
[3] ai.viXra.org:2603.0042 [pdf] submitted on 2026-03-09 02:02:38
Authors: Chaiya Tantisukarom
Comments: 14 Pages.
This paper presents the Prime Gear Geometry (PGG) formulation, a deterministic spectral framework that models prime numbers as discrete mechanical oscillators. By representing the prime sequence as a unit-normalized indicator function ($a_n in {0, 1}$) and applying a band-limited Inverse Fourier Transform (IFT) with a fixed Atomic Tuning Factor ($K=3$), we resolve the individual harmonic identities of adjacent primes. Unlike traditional analytic methods that result in spectral blurring, the PGG formulation ensures the absolute separation of ``Gears'' 2 and 3, maintaining a consistent inter-state valley ($v(2.5) approx -0.566$) across scales ranging from $N=10^2$ to $N=10^6$. We align this framework with Riemann's original observation of the Fourier ``jump'' at prime coordinates, anchoring the discrete forging events to a 0.5 Gyrocenter. Consequently, this paper bypasses the analytic continuation of the Riemann Zeta function, offering a purely mechanical, signal-processing resolution to prime distribution.
Category: Number Theory
[2] ai.viXra.org:2603.0029 [pdf] submitted on 2026-03-06 10:40:23
Authors: Wiroj Homsup
Comments: 3 Pages.
A new Twin prime sieve based on a modified sieve of Sundaram is introduced. It sieves through the set of natural numbers n such that n is not representable in either of the forms 2ij + i + j or2ij + i + j -1 for positive integers i, j.
Category: Number Theory
[1] ai.viXra.org:2603.0014 [pdf] submitted on 2026-03-04 01:15:39
Authors: Chaiya Tantisukarom
Comments: 11 Pages.
This article explores the relationship between the distribution of prime numbers and the zeros of the Riemann Zeta function through the lens of Fourier Analysis. We contrast the ``Natural'' Riemann representation—a discontinuous, jagged summation of discrete frequencies—with the ``Man-made'' Riemann-Siegel $Z(t)$ function. We propose that the Riemann-Siegel remainder term, $R(t)$, acts as a low-pass filter that smooths the underlying digital nature of prime frequencies. This smoothing forces zeros onto the $1/2$ critical line, suggesting that the Riemann Hypothesis may be an artifact of this man-made filtering rather than a fundamental property of the natural prime spectrum.
Category: Number Theory