Number Theory

2511 Submissions

[8] ai.viXra.org:2511.0098 [pdf] submitted on 2025-11-30 22:15:42

On the Diophantine Equation X^2 + K = 2^n: Prime Constraints and Computational Evidence

Authors: Durga Shankar Akodia
Comments: 5 Pages.

u200bWe study integer solutions to the exponential Diophantine equation x^2 + k = 2^n where k and n are primes. We prove that for any solution with n ge 2, x must be odd, and k must be an odd prime. Furthermore, we establish the strict congruence k equiv 7 pmod 8 for all solutions with n ge 3. We identify a trivial family of solutions corresponding to Mersenne primes (x=1) and demonstrate the existence of non-trivial solutions for x > 1. Computational evidence is presented for 9 out of 11 prime values of n le 31, revealing 16 distinct non-trivial solutions. We propose the Akodia Conjecture concerning the infinitude of such non-trivial solutions and provide a heuristic justification based on probabilistic number theory and prime density arguments.
Category: Number Theory

[7] ai.viXra.org:2511.0077 [pdf] replaced on 2025-11-24 23:26:13

Prime-Anchored Fractal Model and Transfer-Matrix Spectrum: Comparison with the Riemann Zeta Function

Authors: Christos Thessalonikios
Comments: 12 Pages.

Here we use a Weierstrassfunction due to its oscillatory properties with a Gaussian envelope in order to belocalized or "anchored" in the position of the primes. Each prime p serves as a localizedfractal anchor generating an oscillatory mode Fp(x), with a delta barrier potential ofthe form VP(x) = Pp gpi δ(x−pi). Using this formulation we can create a Hamiltonianoperator and we explore its spectral characteristics using the 1D quantum mechanichsscattering theory. The zeros of M12(k) tranfer matrix determine a discrete spectrum{kn} that, after global rescaling and some boost of the form t(model) n = αkn + β, alignsperfectly with the 100 imaginary parts of the nontrivial zeros of the Riemann zetafunction on the critical linesn =1 /2 + itnThe correlation of the model and the sn reaches ρ ≈ 1, with a mean absolute deviationbelow 0.011 .
Category: Number Theory

[6] ai.viXra.org:2511.0040 [pdf] replaced on 2026-04-08 06:09:42

On the Impossibility of Divergent Sequences in the Collatz Conjecture

Authors: Dobri Bozhilov
Comments: 3 Pages.

This paper proves that it is impossible to have an infinite sequence of numbers, built according to the rules of the Collatz Conjecture. This is due to the accumulation of unfulfillable requirements for the first member of this sequence when the number of steps becomes infinite.This paper can be seen as a simplified version of the previous paper, entitled "On the impossibility of an infinite chain in The Collatz Conjecture".
Category: Number Theory

[5] ai.viXra.org:2511.0039 [pdf] submitted on 2025-11-13 21:35:47

An Exploratory Analysis of Lehmer Pairs Phase Clustering in Riemann Zeta Zeros

Authors: Gongshan Liu
Comments: 11 Pages.

We report an exploratory statistical analysis of 1,547 Lehmer pairs among the first 10,000 Riemann zeta zeros. Our primary finding is a modest phase clustering pattern: Lehmer pairs show enrichment at phase ϕ ≈ 0.5 in prime-period modulations (observed 29% vs. expected 20%, enrichment 1.45×, p < 10u207b²u2070 after Bonferroni correction for 6 primes). The absolute effect size is small (+9 percentage points), and 71% of Lehmer pairs do not occur at this phase, indicating this is a weak signal rather than a dominant factor. We also observe geometric correlations (R·Δγ = 0.74×, p < 10u207b²u2075) and spatial clustering (79% in clusters), though these may be definitional artifacts. Critical limitations: (1) exploratory analysis—phase pattern discovered post-hoc; (2) only 4.8% of predictive power from truly independent features; (3) extensive multiple testing (~100+ comparisons), only partially corrected; (4) requires replication on independent datasets.
Category: Number Theory

[4] ai.viXra.org:2511.0033 [pdf] submitted on 2025-11-11 20:25:56

Extended Collatz Function as a Markov Chain

Authors: Wiroj Homsup
Comments: 3 Pages. (Note by ai.viXra.org Admin: For the last time, please cite listed scientific references!)

We model the iteration of the extended Collatz function using an infinite-stateMarkov chain. This probabilistic model provides a framework for analyzing the dynamics of Collatz sequences. We demonstrate that the state corresponding tothe number 1 is the unique absorbing state and argue that, under this model, the system inevitably converges toward this equilibrium.
Category: Number Theory

[3] ai.viXra.org:2511.0025 [pdf] submitted on 2025-11-09 03:42:55

Collatz Conjecture model via Infinite Hotel Rearrangement

Authors: Wiroj Homsup
Comments: 2 Pages.

This paper presents an argument based on an analogy with an infinite hotel—where rooms correspond to positive integers—and a rearrangement of guests (also positive integers) following the inverse Collatz-tree rules. The construction aims to demonstrate an inconsistency that challenges the validity of the Collatz conjecture.
Category: Number Theory

[2] ai.viXra.org:2511.0015 [pdf] submitted on 2025-11-06 04:21:13

On the Coprimality of Consecutive Odd Members in Collatz Sequences and Its Implication for Cycle Formation

Authors: Dobri Bozhilov
Comments: 3 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)

The Collatz conjecture posits that for any positive integer n, the sequence generated by the rules n -> n/2 if n is even and n -> 3n + 1 if n is odd always reaches the cycle 4 -> 2 -> 1. This paper presents a novel observation: consecutive odd numbers in a Collatz sequence are coprime, i.e., their greatest common divisor is 1. We prove this property using basic arithmetic and demonstrate that it imposes a significant constraint on the sequence dynamics, particularly for large numbers. Specifically, the requirement of coprimality creates a "pressure" for the next odd number to be smaller than its predecessor, as larger numbers increase the likelihood of common divisors. This tendency toward decrease makes the existence of alternative cycles (distinct from 4 -> 2 -> 1) increasingly improbable, especially for cycles involving extremely large numbers (e.g., exceeding 2^68). Our result supports the conjecture by suggesting that the arithmetic structure of Collatz sequences favors convergence to 1 over the formation of divergent or cyclic sequences.
Category: Number Theory

[1] ai.viXra.org:2511.0002 [pdf] submitted on 2025-11-02 16:23:23

Non-existence of Integer Right Triangles with Both Square Area and Square Perimeter

Authors: Patrick Mcloughlin
Comments: 3 Pages. (Note by ai.viXra.org Admin: Please refrain from making frequent and repeated submissions))

We prove that there is no integer right triangle whose area and perimeter areboth perfect squares. The proof follows directly from the standard parametrisation ofprimitive Pythagorean triples and a short infinite-descent argument.
Category: Number Theory