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[18] viXra:2601.0140 [pdf] submitted on 2026-01-30 01:29:10
Authors: P. N. Seetharaman
Comments: 10 Pages.
This paper presents a straightforward and elementary proof of Fermat's Last Theorem (FLT), asserting that there are no integer solutions to a^n +b^n = c^n for n > 2. Leveraging basic number theory and algebraic manipulations, we offer a concise demonstration aiming to make this fundamental result accessible to a broad mathematical audience.
Category: Number Theory
[17] viXra:2601.0133 [pdf] replaced on 2026-03-10 23:58:09
Authors: Walter A. Kehowski
Comments: 11 Pages. Some improvements in the presentation as well as a new section with an impossibility result.
A power spectral number is a positive integer whose spectral basis consists of only primes and powers. If one searches for power spectral numbers whose spectral sum is also a power, then one finds only five examples. We call these numbers power spectral Pythagorean numbers. The first two examples involve the Pythagorean triples 3,4,5 and 8,15,17. It is shown in this note that these are the only two Pythagorean triples that are power spectral Pythagorean. The other three examples involve the Pell equation.
Category: Number Theory
[16] viXra:2601.0109 [pdf] submitted on 2026-01-23 08:53:31
Authors: Timothy Jones
Comments: 1 Page.
Using tangent lines to the unit circle, we give an argument that shows pi is irrational.
Category: Number Theory
[15] viXra:2601.0103 [pdf] submitted on 2026-01-22 10:15:17
Authors: Rolando Zucchini
Comments: 17 Pages.
Since ancient Greece the possibility of defining natural numbers was considered, but, unlike what happened in Geometry in the Euclid’s Elements, all efforts were in vain. After 2000 years it was the Italian mathematician Giuseppe Peano who was recognized for the historical merit of having provided a rigorous definition of the natural numbers and their properties. His five postulates represent the first well-defined axiomatic foundation of arithmetic. Peano's fifth postulate, known as the Principle of Induction, has provided an indispensable tool in countless mathematical proofs and has enabled significant progress in understanding numbers and their secrets. This paper contains numerous solved exercises on the application of Induction Principle.
Category: Number Theory
[14] viXra:2601.0097 [pdf] submitted on 2026-01-22 21:24:24
Authors: Abdelmajid Ben Hadj Salem
Comments: 7 Pages.
In this paper, assuming that the conjecture c [smaller than] R*2 is true, we give the proof that the explicit abc conjecture of Alan Baker is true and it implies that the abc conjecture is true. We propose the mathematical expression of the constant K(epsilon). Some numerical examples are provided.
Category: Number Theory
[13] viXra:2601.0094 [pdf] replaced on 2026-03-10 06:36:27
Authors: T. Nakashima
Comments: 4 Pages.
Riemann Hypothesis has been the unsolved conjecture for 164 years. This conjecture is the last one of conjectures without proof in "{U}eber die Anzahl der Primzahlen unter einer gegebenen Gr"{o}sse"(B.Riemann). The statement is the real part of the non-trivial zero points of the Riemann Zeta function is 1/2.Very famous and difficult this conjecture has not been solved by many mathematicians for many years. In this paper,I conjecture about the independence (difficulty of proof) of propositions equivalent to the Riemann hypothesis. My position is to discuss the difficulty of proof purely as an intuitive argument.
Category: Number Theory
[12] viXra:2601.0089 [pdf] replaced on 2026-01-27 04:37:39
Authors: Ryan Hackbarth
Comments: 11 Pages. This update includes a more usable primary function, and uses it to forecast the nontrivial zeroes of the Riemann Zeta Function.
Here I present a derivation of an equation whose solution sets are the trivial and nontrivial zeros of the Riemann Zeta Function. I demonstrate how the trivial solutions are directly encoded by integer inputs and how these can be mapped by a symmetry to positive odd integers. I extend this insight to encode the even integers, and map these to the negative odd integers, which provides an explicit connection between particular values of the Riemann Zeta Function which have historical and ongoing research interest. I then extend this symmetry to the nontrivial zeroes, and demonstrate the dependence of the critical line in producing this symmetry. Finally, I note the distribution of the nontrivial zeroes have a correspondence with the distribution of trivial zeroes, and provide a first order approximation of this correspondence.
Category: Number Theory
[11] viXra:2601.0068 [pdf] submitted on 2026-01-16 21:32:20
Authors: Ryan Hackbarth
Comments: 5 Pages. (Note by viXra Admin: Please cite and list scientific references!)
Here I present an equation for the Zeros of the Riemann Zeta Function which connects the distribution of the trivial zeroes with integer inputs to the distribution of the nontrivial zeroes. I demonstrate that this relationship explicitly depends on the critical line where a = ½. I do so in plain language and with replicable calculations, as when I try to write like a mathematician it comes across as inauthentic and bad. Finally, I provide an appendix of calculated solutions.
Category: Number Theory
[10] viXra:2601.0066 [pdf] submitted on 2026-01-15 06:55:09
Authors: J. Kuzmanis
Comments: 9 Pages.
A mathematically simple odd semiprime factorization method is presented.
Category: Number Theory
[9] viXra:2601.0059 [pdf] submitted on 2026-01-14 17:01:00
Authors: Dmitriy S. Tipikin
Comments: 3 Pages.
A famous Fibonacci sequence is forming a simple cycle when sign plus is replaced to minus. A simple proof for any numbers is outlined.
Category: Number Theory
[8] viXra:2601.0052 [pdf] submitted on 2026-01-13 22:50:17
Authors: Youssef Ayyad
Comments: 24 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
Prime numbers have traditionally been studied through the austere lens of arithmetic, yet their deepest structure may be geometric in nature. This work presents a paradigm shift: we construct a toroidal manifold (mathbb{T}^2) where integers are mapped via the phase embedding (Phi(n) = sqrt{n} e^{isqrt{npi}}), transforming discrete divisibility into continuous phase orthogonality. The geometric dust—the area remainder (R(n) = pi n^2 - frac{1}{2}n^3sin(2pi/n))—accumulates into a quantum Hamiltonian (H = -Delta + V) on (mathbb{T}^2). We prove (H) is self-adjoint and its spectrum ({lambda_j}) exhibits Gaussian Unitary Ensemble (GUE) statistics, as verified numerically. Crucially, we propose a textbf{geometric formulation} of the Riemann Hypothesis: we show that, under the assumption of RH, the eigenvalues of (H) are real, bounded below by (frac14), and satisfy the spectral correspondence (lambda_j^{text{(calibrated)}} = frac14 + t_j^2), where (frac12 + it_j) are the non-trivial zeros of (zeta(s)). Numerical verification shows agreement within (0.1%) for the first 50 zeros. The framework reveals primes as ground-state singularities in a resonant field, offering an intuitive geometric foundation for their distribution—not as a proof of RH, but as a novel geometric-spectral formulation of it. For recent developments in geometric approaches to number theory, see Kontorovich and Nakamura (2022), Sarnak (2021), and the survey by Baluyot (2023) on spectral approaches to zeta zeros.
Category: Number Theory
[7] viXra:2601.0050 [pdf] submitted on 2026-01-13 01:28:26
Authors: Debasis Biswas
Comments: 03 Pages.
In this paper Polya equivalent of Riemann Hypothesis is proved from Complex analytic expression of Riemann Xi function.
Category: Number Theory
[6] viXra:2601.0046 [pdf] submitted on 2026-01-12 20:50:18
Authors: Youssouf Ouédraogo
Comments: 21 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org) Creative Commons Attribution 4.0 International
This paper proposes a new structural approach to the study of consecutive prime numbers based on a quadratic relation linking three successive primes. A stability ratio is introduced and shown to converge asymptotically to unity using explicit bounds for the k-th prime number. This convergence induces a constraint on the local variation of prime gaps, leading to an asymptotic smoothness law for their relative fluctuations. The analysis is fully deterministic and avoids heuristic arguments based on average asymptotic. Numerical validations using verified large prime datasets confirm the theoretical predictions and illustrate the progressive regularization of local gap variations as the prime index increases.
Category: Number Theory
[5] viXra:2601.0039 [pdf] submitted on 2026-01-11 01:30:39
Authors: Ryan Hackbarth
Comments: 2 Pages.
In this paper, I present a formula for the zeroes of the Riemann Zeta Function and highlight their dependence on a rational integer ratio. I connect these ratios with a hyperbola reminiscent of Pell’s equation which approximates pi and provide a table of calculated ratios and their corresponding Zero. Finally, I demonstrate the requirement of the critical line at ½ in producing these integer approximations.
Category: Number Theory
[4] viXra:2601.0024 [pdf] submitted on 2026-01-06 14:29:32
Authors: Brian Scannell
Comments: 25 Pages.
To support intuitive understanding of Fermat’s Last Theorem, this paper presents a simple visualisation based on a defined normalised Fermat plot and shows that rational directions arising from succession t Pythagorean triples—with a fixed hypotenuse gap—become automatically irrational beyond a finite point, explaining why no Fermat type integer solutions can occur along these directions.
Category: Number Theory
[3] viXra:2601.0020 [pdf] submitted on 2026-01-05 20:31:18
Authors: Silvio Gabbianelli
Comments: 7 Pages. (Note by viXra Admin: Please cite and list scientific references)
By observing the relative positions of odd composite numbers in the set of odd natural numbers up to a given n, the positions of the prime numbers can be logically derived by subtraction. Not only that, but a linear, albeit parametric, function can also be deduced that can provide all and only the odd composite natural numbers up to n, and therefore all the prime numbers up to n. This allows us toformulate the conjecture that the set of prime numbers (except 2) is the well-ordered complementary set of odd composite numbers. This ordering can also be seen using the Cartesian line y = 2x + 1. Other lines and different numberings can highlight other possible properties of prime numbers.
Category: Number Theory
[2] viXra:2601.0016 [pdf] submitted on 2026-01-04 14:36:58
Authors: Giovanni Di Savino
Comments: 5 Pages.
Thales measured the height of the inaccessible pyramid and the distance of the unreachable ship from the harbor, demonstrating that anything that can be plotted on a plane can be measured; Euclid, with the product of known prime numbers, continually generates new primes and demonstrated that prime numbers are infinite; Peano, with the second of his five axioms, affirmed that for every natural number there exists a successor number +1. We will never be able to claim to have developed Euclid's inaccessible primes or Peano's unattainable number, but twin primes are two of the infinite primes, one of which is a successor number +2 of the other prime, and the sum of the two primes is always a number 6n; by representing even numbers in the form 6n or 6n±2 and odd numbers in the form 6n±1 or 6n±2±1, we can demonstrate that Euclid's inaccessible primes and Peano's unattainable successor number exist. All prime numbers, all twin primes, all Mersenne primes which are the sum of numbers in double proportion and generate the even perfect numbers, all odd numbers 3n of the Collatz algorithm whose successor +1 is a power 2^n_even which when halved is 2^(n-1) and ends at 2^0 = 1 and all even numbers and all odd numbers which are the sum of 2 or 3 primes, all exist and, even if they will never be known, the final digit of the prime numbers and of the successor number which can be a prime or composite number is known.
Category: Number Theory
[1] viXra:2601.0001 [pdf] replaced on 2026-01-25 00:27:26
Authors: Keshava Prasad Halemane
Comments: 16 Pages. 1 Table
This research report presents the Collatz-Hasse-Syracuse-Ulam-Kakutani (CHSUK) Theorem, which asserts the convergence of the Collatz Sequence to the trivial cycle, thus proving the Collatz Conjecture, which has been a long-standing unsolved problem. The proof is based on the bijective isomorphism established between the set of positive integers and a carefully designed system with a hierarchy (arborescence) of binary-exponential-ladders defined on the set of positive odd numbers.
Category: Number Theory
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