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[4] viXra:2601.0138 [pdf] submitted on 2026-01-30 01:04:02
Authors: Edgar Valdebenito
Comments: 4 Pages.
This note is about a specific value of Lambert's W function.
Category: General Mathematics
[3] viXra:2601.0126 [pdf] replaced on 2026-03-23 02:16:35
Authors: Zhi Li, Hua Li
Comments: 6 Pages.
In mathematics, real numbers can be represented by points on a straight line called the number line, which includes a point called the origin, the direction of number growth, and a unit length. It is generally assumed that there is a one-to-one correspondence between real numbers and points on the number line, with the position of a point determining the size and order of the numbers. This essentially assumes that all real numbers have a definite position on the number line, and that there is a definite order between any two real numbers.This paper shows that there are real numbers with uncertain positions, and that all real numbers do not lie on the number line of the same dimension. The number line is composed of discrete points, which are "pure numbers"—that is, only pure numbers exist on the number line, while non-pure numbers exist in "empty space." Therefore, there is a logical contradiction between the continuity of real numbers and the real number line; the real number line is an incomplete and imperfect conception for representing real numbers. This paper gives the definition of a pure number and the relationship between its cardinality and the natural cardinality.These results verify the viewpoint of quantum theory in physics, namely that the straight line on the "macroscopic" number line is composed of "microscopic" discrete and discontinuous points.
Category: General Mathematics
[2] viXra:2601.0101 [pdf] submitted on 2026-01-22 21:26:54
Authors: Juan Moreno Borrallo
Comments: 19 Pages. (Note by viXra Admin: For the last time, please submit article written with AI assistance to ai.viXra.org!)
The arithmetic of the integers is governed by two fundamental operations, addition and multiplication, whose interaction lies at the core of many deep problems in number theory. While multiplication preserves prime factorization in a rigid and conservative manner, addition typically destroys multiplicative structure and generates new prime content.In this work, we develop a unified structural framework that explains this asymmetry through spectral and operator-theoretic principles. By embedding the integers into a Hilbert space, we show that multiplication acts as a diagonal, layer-preserving operator in the prime spectral basis, whereas addition acts as a non-local, mixing operator driven by carry propagation. This spectral incompatibility leads to an arithmetic uncertainty principle, forbidding simultaneous localization in additive and multiplicative bases.Building on this structure, we introduce additive innovation as a quantitative measure of the new prime information created by a sum. We prove that the only obstruction to innovation arises from smoothness and $S$-unit phenomena in the coprime core. Using classical results on smooth numbers, we show that additive innovation is typically large, yielding unconditional abc-type inequalities in density.Finally, we develop an information-theoretic perspective, showing that addition produces entropy across prime scales while multiplication remains information-preserving. These results provide a structural explanation for the sum-product phenomenon and reframe classical problems as manifestations of the intrinsic incompatibility between additive and multiplicative spectral structures.
Category: General Mathematics
[1] viXra:2601.0096 [pdf] submitted on 2026-01-22 21:20:43
Authors: Mattia Furlin
Comments: 5 Pages.
In this short article, we will discuss a card game, from now on namely Solitaire modulo 3. After having described how it works, through a probabilistic calculation, we will arrive at determining the probability of victory. In particular, we will use the rook polynomials, which will allow us to finally obtain a closed form for calculating the probability of winning at Solitaire modulo 3. Finally, we will study the case where the number of cards in play is much more greater than the number of constraints present in the game format. Under this assumption, the Solitaire modulo 3 mechanism becomes asymptotically equivalent to a binomial distribution.
Category: General Mathematics
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