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[6] viXra:2602.0102 [pdf] submitted on 2026-02-19 20:14:58
Authors: Tajmul Khan
Comments: 13 Pages. (Note by viXra Admin: Please cite listed scientific references)
We present a new analytic framework for generating infinite series representations of pi. Using the digamma reflection formula and a modulus-q expansion strategy, we construct families of convergent pi-series. In particular, we derive a novel modulus-5 weighted series (Khan's pi-series), which converges absolutely and exhibits faster convergence than the classical Leibniz series. The approach generalizes naturally to arbitrary moduli q >= 3, yielding entire families of pi-series with adjustable convergence behaviour.
Category: General Mathematics
[5] viXra:2602.0091 [pdf] submitted on 2026-02-18 19:47:27
Authors: Binay Krishna Maity
Comments: 3 Pages.
We have two straight lines in a graph. We need to determine if these straight lines will meet each other or remain parallel if we extend these lines. This paper helps us to determine this question. If we square a certain type of polynomial (x^pn - x^p(n-1) -....... -x^p1-1) and take the coefficient of x of this square on the x axis and the power of x on the y axis and if we make the graph, the spectra that will be created, consider the initial and final part of the spectra as two straight lines, then those two straight lines will meet each other or be parallel, it will depend on the n and p of this polynomial. That is, on the length and step of the polynomial.
Category: General Mathematics
[4] viXra:2602.0085 [pdf] submitted on 2026-02-17 00:31:42
Authors: Binay Krishna Maity
Comments: 4 Pages. (Note by viXra Admin: Please cite and list scientific references!)
A very common term in mathematics is a^2 + b^2 = c^2. This equation has been discussed since many years ago. This equation is called the Pythagorean Theorem. And for a, b and c as positive inintegers a, b and c are called Pythagorean triples. For example, 3, 4, 5 is a Pythagorean triple. Because, 3^2+ 4^2 = 5^2. Some more examples are (5,12,13), (9,12,15), and (12,16,20) etc. Any number has more than one Pythagorean triple. For example, with 12 numbers (5,12,13), (9,12,15), (12,16,20) and (12, 35, 37) these four Pythagorean triples are obtained. Now I am given a positive integer number and I have to show how many Pythagorean triples can be found with that number? There may have some solutions in the mathematics. This paper provides an alternative solution to evaluate these Pythagorean triplets for any given number.
Category: General Mathematics
[3] viXra:2602.0084 [pdf] submitted on 2026-02-17 00:24:20
Authors: Arthur Shevenyonov
Comments: 6 Pages.
The present paper proposes a most parsimonious scheme to arrive at solutions for polynomial algebraic (or ODE) equations of an arbitrary degree (order). The former option treats the polynomial as an implied characteristic of a difference/recurrent equation (dubbed an AlD, or algebraic-to-difference path). The latter (i.e. ODE) domain, rather than building on the more trivial differential-difference parallelism, embarks on first reducing ODE to algebraic (while making use of, say, Mikusinski-style E-operators) then applying the above procedure (what amounts to a DARF/DARE, or differential to algebraic to recurrent/functional equation path).
Category: General Mathematics
[2] viXra:2602.0082 [pdf] submitted on 2026-02-15 10:06:26
Authors: Mieczyslaw Szyszkowicz
Comments: 5 Pages.
Archimedes used the perimeter of inscribed and circumscribed regular polygons to obtain lower and upper bounds of the number pi. He started with two regular hexagons and he doubled their sides from 6 to 12, 24, 48, until 96. Applying the perimeters of 96 side regular polygons, Archimedes obtained the bounds for the number pi: 3+10/71<pi<3+1/7. His algorithm can be executed as a recurrence formula called the Borchardt-Pfaff-Schwab method. Dörrie proposed an improvement of this algorithm to produce narrower interval which encapsulates pi. Here a linear combination of the bounds is realized to obtain an improved accuracy. Many other linear combinations are presented to approximate this mathematical constant.
Category: General Mathematics
[1] viXra:2602.0052 [pdf] submitted on 2026-02-07 01:20:20
Authors: Felix M Lev
Comments: 17 Pages.
As shown by Gödel and other mathematicians, foundational problems of classical mathematics (CM) arise because this theory involves the entire infinite set of natural numbers. Therefore, CM must be modified in some way. A problem discussed in a wide literature is how mathematics should be treated: (1) as a purely abstract discipline, independent of nature; or (2) as a discipline that must ultimately describe nature. Most physicists accept only viewpoint (2), while many mathematicians and philosophers adopt viewpoint (1). However, currently approach (1) did not solve the problem of how CM should be modified, and quantum theory (QT) is considered to be the most general theory for describing nature. Therefore, CM must be modified so that it correctly describes QT. As shown in our publications, finite mathematics (FM) satisfies this condition. It involves a finite ring $R_p=(0, 1, ...p-1)$ where addition, subtraction, and multiplication are performed modulo $p$. FM does not contain any foundational problems and is a more general theory than CM: the latter is a degenerate special case of the former in the limit $ptoinfty$. The purpose of this paper is to provide a brief overview of our results to make them understandable to a wide audience of mathematicians and physicists.
Category: General Mathematics
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