Indian Journal of Advanced Mathematics (IJAM)
type: channel
description: The Indian Journal of Advanced Mathematics (IJAM) has ISSN 2582-8932 (online), an open-access, peer-reviewed, periodical half-yearly international journal, which is published by Lattice Science Publication (LSP) in April and October. Authors can electronically submit articles throughout the year using the Article Submission System. Authors can format their article either in (i) a single-column format or (ii) as per the journal template. The submitted articles should not have been previously published or are currently under consideration for publication elsewhere. The journal does not accept brief or short notes for publication. The editors retain the right to reject any articles that lack quality or originality without sending them for review. All articles must fall within the journal’s scope and will undergo a double-anonymized peer-review process. Authors must confirm that they have read and understood the content of their submitted article and ensure that it meets acceptable English grammar and usage standards. To help with the proofreading process, authors can use tools like Grammarly or similar applications. As an open-access journal, authors must pay an Article Processing Charge (APC) to publish their articles and retain copyright. Additionally, authors should familiarise themselves with the editorial and publishing policies of the journal.
publisher: Lattice Science Publication (LSP)
language: en
publication name: Indian Journal of Advanced Mathematics (IJAM)
issn: 2582-8932
copyright: CC-BY-NC-ND 4.0
items: In Search of an Elementary Proof for Fermat's Last Theorem
items: A Comprehensible Proof for Fermat's Last Theorem
items: Solution of Brocard’s Problem
items: An MDS Code Generated from a Three-Term Exponential Sum
items: Analyzing the Chinese Remainder Theorem to Find Solutions Efficiently
items: Riemann Hypothesis: The Most Definitive and Comprehensive Disproof Ever Constructed
items: On Group A-Cordial Labelling of Double Uniform (t1l1,t2l2) Ply
items: Fractional form of Jensen’s Inequality for the Exponential Integral Function with Applications to Modeling Utility
items: Goldbach Conjecture: The Most Definitive and Comprehensive Disproof Ever Constructed
items: Unveiling an Elementary Proof for Fermat's Last Theorem
items: A General Formula for the Probability of Winning in Sequential Turn-Based Games
items: The Time-Long Misconceptions in Classical Mathematics
items: Non-Trivial Zeros of the Riemann Zeta Function as Zero Displacement Vectors
items: Study and Consequences of the Ⓢ Function on the Riemann Hypothesis
items: Confirm Goldfeld Conjectured for Infinitely Many Elliptic Curves
items: Every Sum of Two Positive Integers Has Either a Trivial or a Non-Trivial Common Factor
items: Properties of Objects and Their Transformation on the Plane
items: An Elementary Procedure in the Proof of Fermat's Last Theorem
items: Re-Evaluating the Value of π and Emerging New Concepts for Measuring Areas and Volumes Based on Translations and Rotations of Straight Lines, Planes, and Circles
items: Hybrid Mean Value of General Quartic Gauss Sums and Three-Term Exponential Sums
items: An Index-Jets Framework for Irrationality: from Sturm-Liouville Operators to the AGM
items: Numerical and Analytical Method based Mathematical Approach to Estimate Water Pollution over 3-D Region
items: Complex Dynamics of an Aquatic Tri-Trophic Food Chain System with Holling Type IV Functional Response
items: Comment on ‘The Triple Product Rule and the Subtleties of the Mathematical Tools Used in Thermodynamics’ by Felipe Américo Reyes Navarro and Rafael Edgardo Carlos Reye
items: A Comparative Study on the Effects of Vertical and Horizontal Magnetic Fields on Double Diffusive Convection with Soret Effect in Couple-Stressed Nanofluids
items: Completed Functor S-1() of the Localization Functor S-1(), Isomorphism and Adjunction
items: Study of Some Degree Five Identities of Type (4,1)
items: Existence and Uniqueness of Weak Solutions to the Fractional Navier-Stokes Equations in Turbulent Flows: A Study of Boundedness and Stability
items: An Innovative Approach to Find Remainder
items: Some Results on Specificity of Possibility Distributions

In Search of an Elementary Proof for Fermat's Last Theorem
type: item
description: Fermat’s Last Theorem states that the equation x n + y n = z n has no solution for x, y and z as positive integers, where n is any positive integer > 2. Taking the proofs of Fermat and Euler for the exponents n = 4 and n = 3, it would suffice to prove the theorem for the exponent n = p, where p is any prime > 3. We hypothesise that r, s, and t are positive integers satisfying the equation rp + sp = tp, and we establish a contradiction in this proof. We include another Auxiliary equation, x³ + y³ = z³, and connect these two equations by using the transformation equations. By solving the transformation equation, we prove that rst = 0, thus demonstrating that only a trivial solution exists in the central equation, r p + s p = t p .
subject: Articles
  topic: section
subject: Transformation Equations To Two Fermat's Equations
  topic: keywords
subject: Mathematics Subject Classification 2010
  topic: keywords
subject: 11A–XX
  topic: keywords
creator: P.N. Seetharaman
rights: Copyright (c) 2026 P.N. Seetharaman (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-04-24
publication date: 2026-04-24
starting page: 35
ending page: 39
doi: 10.54105/ijam.A1190.04010424/

A Comprehensible Proof for Fermat's Last Theorem
type: item
description: Fermat’s Last Theorem states that it is impossible to find positive integers A, B and C satisfying the equation An + Bn = Cn where n is any integer > 2. Taking the proofs of Fermat for the index n = 4, and Euler for n = 3, it is sufficient to prove the theorem for n = p, any prime > 3. We hypothesise that all r, s, and t are non-zero integers in the equation rp + sp = tp and establish a contradiction. To support the proof in the above equation, we have another equation: x³ + y³ = z³. Without loss of generality, we assume that both x and y are non-zero integers, z³ is a non-zero integer, and z and z² are irrational. We transform the above two equations using parameters, incorporating the Ramanujan-Nagell equation. Solving the transformed equations, we prove the theorem.
subject: Articles
  topic: section
subject: Transformed Fermat’s Equations through Parameters
  topic: keywords
subject: 2010 Mathematics Subject Classification 2010
  topic: keywords
subject: 11A–XX
  topic: keywords
creator: P. N Seetharaman
rights: Copyright (c) 2026 P. N Seetharaman (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-04-24
publication date: 2026-04-24
starting page: 29
ending page: 34
doi: 10.54105/ijam.A1181.04010424

Solution of Brocard’s Problem
type: item
description: Brocard's problem is the solution of the equation,n! + 1=m2, where m and n are natural numbers. So far, only three solutions have been found, namely (n, m) = (4, 5), (5, 11), and (7, 71). The purpose of this paper is to show that there are no other solutions. Firstly, it will be shown that if (n,m) is to be a solution to Brocard's problem, then n! = 4AB, where A is even, B is odd, and |A – B| = 1. If n is even (n = 2x) and > 4, it will be shown that necessarily A = (2x)‼ 4y and B = y (2x-1)‼, for some odd y > 1. Next, it will be shown that x < 2y, and this leads to an inequality in x [namely, (x(2x − 1)‼ +- 1) 2 − 1 − (2x)! < 0], for which there is no solution when x ≥ 3. If n is odd, a similar procedure applies.
subject: Articles
  topic: section
subject: Brocard's Problem
  topic: keywords
subject: Diophantine Equation
  topic: keywords
subject: Brown Numbers
  topic: keywords
creator: Mohamed Imteaz Karimullah
rights: Copyright (c) 2026 Mohamed Imteaz Karimullah (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-04-24
publication date: 2026-04-24
starting page: 25
ending page: 28
doi: 10.54105/ijam.B1174.04010424/

An MDS Code Generated from a Three-Term Exponential Sum
type: item
description: In this paper, I investigate the method for constructing Maximum Distance Separable (MDS) codes using exponential sums. By utilizing the properties of the trace function over finite fields, I explicitly calculate linear codes associated with certain three-term exponential sums. The parameters, weight distributions, and distance properties of these codes are analyzed in detail. Furthermore, we analyse the syndrome-decoding process for these codes.
subject: Articles
  topic: section
subject: Three-Term Exponential Sum
  topic: keywords
subject: Linear Code
  topic: keywords
subject: Syndrome Decoding
  topic: keywords
subject: MDS Code
  topic: keywords
subject: MSC: 11 LO5
  topic: keywords
creator: Dr. Shikha Singh
rights: Copyright (c) 2026 Dr. Shikha Singh (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-04-30
publication date: 2026-04-30
starting page: 57
ending page: 61
doi: 10.54105/ijam.A1241.06010426

Analyzing the Chinese Remainder Theorem to Find Solutions Efficiently
type: item
description: In this paper, we explain the logic for simultaneously solving linear congruences using the Chinese Remainder Theorem and apply it to develop an easy method for finding the solution.
subject: Articles
  topic: section
subject: Simultaneous Linear Congruences
  topic: keywords
subject: Chinese Remainder Theorem
  topic: keywords
creator: Dr. Aditi Sunil Phadke
rights: Copyright (c) 2026 Dr. Aditi Sunil Phadke (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-04-30
publication date: 2026-04-30
starting page: 55
ending page: 56
doi: 10.54105/ijam.A1244.06010426

Riemann Hypothesis: The Most Definitive and Comprehensive Disproof Ever Constructed
type: item
description: For over 160 years, the Riemann Hypothesis has stood as a cornerstone of modern mathematics: profound, elegant, and elusive. Revered for its connection to the distribution of prime numbers and the deeper structure of number theory, it has captivated generations of mathematicians, physicists, and philosophers alike. Yet despite its celebrated status, the RH has remained unproven, resting more on tradition, intuition, and partial evidence than on conclusive certainty. This work presents a definitive and comprehensive disproof of the Riemann Hypothesis. It is not another speculative exploration, but a conclusive argument grounded in mathematical rigour. Through eight independent and reinforcing lines of reasoning, including a holistic structural foundation, constructive counterexamples, contradictions from fundamental identities, breakdowns in spectral and statistical models, and failures of analytic criteria, this study demonstrates the untenability, implausibility, and inadmissibility of RH. Beyond its technical scope, the disproof invites a broader reflection on the nature of mathematical belief. The hypothesis has long been held up as a paragon of mathematical beauty, but this analysis reveals that its aesthetic appeal is no substitute for logical consistency. Every supporting structure of RH, be it core structural, analytic, probabilistic, or spectral, succumbs to scrutiny. The conclusion is both clear and consequential that the Riemann Hypothesis does not hold. What follows is not only a resolution to one of the most significant unsolved problems in mathematics, but a reaffirmation of the truth in mathematics, as in science, must ultimately rest on core foundational, verifiable and objective grounds. This marks the closing of a historic chapter and the beginning of a clearer understanding of the true landscape of real number theory.
subject: Articles
  topic: section
subject: RH. Beyond Its Technical Scope
  topic: keywords
subject: Every Supporting
  topic: keywords
subject: Structure
  topic: keywords
creator: Sandeep S. Jaiswal
rights: Copyright (c) 2026 Sandeep S. Jaiswal (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-04-30
publication date: 2026-04-30
starting page: 49
ending page: 54
doi: 10.54105/ijam.B1203.06010426

On Group A-Cordial Labelling of Double Uniform (t1l1,t2l2) Ply
type: item
description: Let A denote the multiplicative group {1,−1, i, −i}. In this paper, we define the notion of double uniform (t1l1,t2l2)-ply and prove that it is a group A-cordial with each path of length at least 5 by explicitly giving labellings for all the possible cases that arise. Mathematics Subject Classification [2020]: Primary 05C78
subject: Articles
  topic: section
subject: t-ply
  topic: keywords
subject: Uniform t-ply
  topic: keywords
subject: Double t-ply
  topic: keywords
subject: Double Uniform t-ply
  topic: keywords
subject: Group A-Cordial Labelling
  topic: keywords
creator: Dr. Samina Boxwala
creator: Dr. Aditi S. Phadke
creator: Pramod N. Shinde
creator: Dr. Nilesh Mundlik
rights: Copyright (c) 2026 Dr. Samina Boxwala, Dr. Aditi S. Phadke, Pramod N. Shinde, Dr. Nilesh Mundlik (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-04-30
publication date: 2026-04-30
starting page: 44
ending page: 48
doi: 10.54105/ijam.A1242.06010426

Fractional form of Jensen’s Inequality for the Exponential Integral Function with Applications to Modeling Utility
type: item
description: Jensen’s inequality is a fundamental result in probability and analysis, that offers simple boundsforthe convex function applied to a random variable. However, the classical form of this process does not include memory effects, which are very important in some physical and financial systems with long-range dependence. In this paper, we introduce a fractional-order generalisation of Jensen’s inequality involving memory effects that can be accounted for by means of fractional calculus. We concentrate on the exponential integral function Ei(x) because of its wide use. For a random variable Xwith mean µsupported on [1, ∞), and for every fractional-order α ∈ (0,1) we show the strict inequality Ei(µ) ≤ E[Ei(X)] − M(α), where the new quantity M(α) is a memory correction defined in terms of Riemann– Liouville fractional integrals of order 1 − αof the function e t /t. The correction term provides a whole family of bounds, controlled by the parameter α ∈ (0, 1], and depending on the specific behavior of X, as α→ 0+ , the bound reduces to Jensen’s gap, while for α → 1 —, the right-hand side approaches a new non-zero and pathdependent bound. The inequality is strict for non-degenerate X. (When M(α) ≥ 0, the bound is two-sided, Ei(µ) ≤ E[Ei(X)] − M(α) ≤ E[Ei(X)].) The proof is based on order-monotonicity properties of fractional integrals. An equivalent formulation of the result in terms of Caputo derivatives is also given. An illustrative interpretation is discussed in the context of economic utility, where the resulting bounds may be viewed as capturing nonlocal averaging effects in convex (risk-seeking) utility evaluations.
subject: Articles
  topic: section
subject: Jensen’s Inequality
  topic: keywords
subject: Fractional Calculus
  topic: keywords
subject: Caputo Derivative
  topic: keywords
subject: Exponential Integral
  topic: keywords
subject: Convex Analysis
  topic: keywords
subject: Recursive Utility
  topic: keywords
subject: Memory Effects
  topic: keywords
subject: Risk Assessment
  topic: keywords
creator: Dr. Issah Imoro
creator: Dr. Ahmed Yakubu
creator: Dr. Stephen Napio Ajega-Akem
rights: Copyright (c) 2026 Dr. Issah Imoro, Dr. Ahmed Yakubu, Dr. Stephen Napio Ajega-Akem (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-04-30
publication date: 2026-04-30
starting page: 36
ending page: 43
doi: 10.54105/ijam.A1234.06010426

Goldbach Conjecture: The Most Definitive and Comprehensive Disproof Ever Constructed
type: item
description: This paper presents the first complete and definitive disproof of Goldbach's Conjecture, executed through a synthesis of deep mathematical insight, theoretical innovation, and structural rigour. Across a broad spectrum of modern mathematics, this work reveals foundational contradictions that undermine the conjecture's claim to universality. The result is a clear and categorical conclusion that Goldbach's Conjecture, while numerically resilient, collapses under formal scrutiny. This work not only resolves a centuries-old enigma but redefines the philosophical foundation of additive number theory. It proactively challenges the mathematical community to distinguish between empirical tradition and provable truth, and sets a new standard for resolving longstanding conjectures. In doing so, it transforms the landscape of number theory and establishes a model for multidisciplinary proof that enables future breakthroughs. This is not merely a mathematical achievement but a historic turning point as the era of Goldbach concludes, and the era of mathematical evolution begins.
subject: Articles
  topic: section
subject: Deep Mathematical Insight
  topic: keywords
subject: Theoretical Innovation
  topic: keywords
subject: Structural Rigour
  topic: keywords
subject: Numerically Resilient
  topic: keywords
creator: Sandeep S. Jaiswal
rights: Copyright (c) 2026 Sandeep S. Jaiswal (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-04-30
publication date: 2026-04-30
starting page: 31
ending page: 35
doi: 10.54105/ijam.B1204.06010426

Unveiling an Elementary Proof for Fermat's Last Theorem
type: item
description: 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 that makes this fundamental result accessible to a broad mathematical audience.
subject: Articles
  topic: section
subject: Transformation Equations
  topic: keywords
subject: Fermat's Equations
  topic: keywords
subject: 2020 Mathematics Subject Classification (MSC) Code: 11A-XX
  topic: keywords
creator: Palamadai Narayanan Seetharaman
rights: Copyright (c) 2026 Palamadai Narayanan Seetharaman (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-04-30
publication date: 2026-04-30
starting page: 24
ending page: 30
doi: 10.54105/ijam.A1239.06010426

A General Formula for the Probability of Winning in Sequential Turn-Based Games
type: item
description: This paper presents a general formula for calculating a player's probability of winning in a sequential, turn-based game with a constant success probability per trial. The problem extends the classical two-player probability models of dice tossing or coin flipping to an arbitrary number of n players. A compact proof based on the summation of a geometric series is provided, and examples demonstrate the correctness and applicability of the result. This formulation can serve as an educational tool for understanding probabilistic reasoning, sequences, and infinite series.
subject: Articles
  topic: section
subject: Probability
  topic: keywords
subject: Successive Trials
  topic: keywords
subject: Random Experiments
  topic: keywords
subject: Sequential Games
  topic: keywords
subject: Geometric Series
  topic: keywords
subject: Binomial Distribution
  topic: keywords
creator: Sakshya Vardhan Mishra
rights: Copyright (c) 2026 Sakshya Vardhan Mishra (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-04-30
publication date: 2026-04-30
starting page: 22
ending page: 23
doi: 10.54105/ijam.A1226.06010426

The Time-Long Misconceptions in Classical Mathematics
type: item
description: Many foundational rules in elementary and advanced mathematics are accepted as axiomatic, despite being introduced through heuristic or pedagogical arguments rather than strict logical or physical justification. This work undertakes a systematic analytical re-examination of several such widely accepted mathematical conventions to identify internal inconsistencies and clarify their conceptual basis. The study begins by reanalysing the four basic arithmetic operations—addition, subtraction, multiplication, and division—by reducing multiplication and division to repeated addition and subtraction. Within this framework, the conventional sign rules governing the multiplication of negative quantities are critically examined. It is shown that while the outcomes of these rules are operationally consistent, the standard logical justifications commonly provided are incomplete or internally inconsistent when interpreted in terms of direction, orientation, and repetition. Using geometric constructions and physically motivated examples, the work further examines the interpretation of signed quantities, demonstrating that positive and negative values naturally encode directionality rather than intrinsic magnitude. This perspective is extended to areas and volumes, which are typically assumed to be strictly non-negative. The analysis shows that signed areas and volumes can be meaningfully interpreted within a consistent mathematical–physical framework, thereby questioning one of the key assumptions underlying the introduction of imaginary quantities. The paper also revisits the treatment of fractional quantities, squaring operations, and dimensional comparisons, arguing that several commonly cited “paradoxes” arise from conflating quantities of different dimensions or from scaledependent representations rather than from inherent mathematical necessity. Additionally, division by zero and infinity is reinterpreted through the lens of repeated subtraction, yielding a physically intui…
subject: Articles
  topic: section
subject: Mathematical Misconceptions
  topic: keywords
subject: Negative Numbers
  topic: keywords
subject: Imaginary Numbers
  topic: keywords
subject: Division by Zero & Exponential Constant (e)
  topic: keywords
creator: Nishant Sahdev
creator: Dr. Chinmoy Bhattacharya
rights: Copyright (c) 2026 Nishant Sahdev, Dr. Chinmoy Bhattacharya (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-04-30
publication date: 2026-04-30
starting page: 11
ending page: 21
doi: 10.54105/ijam.A1236.06010426

Non-Trivial Zeros of the Riemann Zeta Function as Zero Displacement Vectors
type: item
description: In this paper, we show that a non-trivial zero of the Riemann zeta function occurs only when the complex number s = a/b + it, with a, b, t element of R and i² = -1 can be interpreted as a vector plus its inverse yielding zero displacement. We prove that for such a zero displacement to occur, the total distance covered by the vector and itsinverse must equal one unit, forcing the fundamental part of s to be 1 2. We further show that no other fraction in the critical strip possesses this property. Consequently, no other fundamental part can host non-trivial zeros, thereby settling the Riemann Hypothesis.
subject: Articles
  topic: section
subject: Riemann Zeta Function
  topic: keywords
subject: Equal One Unit
  topic: keywords
subject: Fundamental Part
  topic: keywords
creator: Joseph Kongani Wamukoya
rights: Copyright (c) 2026 Joseph Kongani Wamukoya (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-04-30
publication date: 2026-04-30
starting page: 9
ending page: 10
doi: 10.54105/ijam.A1233.06010426

Study and Consequences of the Ⓢ Function on the Riemann Hypothesis
type: item
description: I will study the Sghiar’s function Ⓢ:(X,z)⟼∏_(p∈P)1/(1-X/p^z ), P the set of prime numbers. Which is an extension of the Riemann zeta function. The classical form of the Riemann zeta function and its Euler product are well known in analytic number theory, and I will show that : ζ(s)=0 and Re(s)>1/2⇒Ⓢ=0. We deduce the proof of the Riemann Hypothesis. MSC code: 11M26 ; 97F60 ; 32A10
subject: Articles
  topic: section
subject: Prime Number
  topic: keywords
subject: Holomorphic Function
  topic: keywords
subject: Riemann Hypothesis
  topic: keywords
creator: Mohamed Sghiar
rights: Copyright (c) 2026 Mohamed Sghiar (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-04-30
publication date: 2026-04-30
starting page: 5
ending page: 8
doi: 10.54105/ijam.A1231.06010426

Confirm Goldfeld Conjectured for Infinitely Many Elliptic Curves
type: item
description: Goldfeld conjectured: "a positive proportion of quadratic twists of an elliptic curve E/Q have an analytic rank of 1. In this work, we confirm this assertion for infinitely many elliptic curves.
subject: Articles
  topic: section
subject: Elliptic Curves
  topic: keywords
subject: Goldfeld Conjectured
  topic: keywords
subject: Analytic Rank
  topic: keywords
subject: Optimal Elliptic Curve
  topic: keywords
subject: Modular Curve and Binary Goldbach Problem
  topic: keywords
creator: Dr. Safwan Aouira
creator: Dr. Hasan Sankari
creator: Ahmad Abdo
rights: Copyright (c) 2026 Dr. Safwan Aouira, Dr. Hasan Sankari, Ahmad Abdo (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-04-30
publication date: 2026-04-30
starting page: 3
ending page: 4
doi: 10.54105/ijam.A1230.06010426

Every Sum of Two Positive Integers Has Either a Trivial or a Non-Trivial Common Factor
type: item
description: This paper investigates the arithmetic structure of exponential Diophantine equations of the form A x + B y= k n , where A, B, k, x, y, n ∈ Z + . Classical treatments such as the Beal Conjecture [1] and Fermat’s Last Theorem (FLT) [2] restrict attention to exponents greater than two, leaving open the structural behavior of the equation for n = 1 and n = 2. This manuscript provides a unified framework addressing all positive integer exponents. A central theorem establishes that each term A x , B y , and k n can be expressed as the sum of an arithmetic sequence whose number of terms and average term are positive integers, provided the equation has a trivial or non-trivial common factor. This elasticity property of k n is derived through Gauss’s method for summing arithmetic progressions. The case n = 2 recovers the classical identity for k 2 as the sum of the first k odd integers [3], revealing Pythagoras’ theorem as a special instance of the general framework. For exponents exceeding two, if gcd(A, B,k) = 1, the arithmetic structure collapses, aligning with the Beal Conjecture as it is presented in the literature as a generalization of FLT [1]. The results demonstrate a consistent theory for all positive integer exponents and show that every sum of two positive integers has either a trivial or a non-trivial common factor.
subject: Articles
  topic: section
subject: Beal Conjecture
  topic: keywords
subject: Arithmetic Sequences
  topic: keywords
subject: Exponential Diophantine Equations
  topic: keywords
subject: Fermat’s Last Theorem
  topic: keywords
subject: Pythagoras
  topic: keywords
creator: Joseph Kongani Wamukoya
rights: Copyright (c) 2026 Joseph Kongani Wamukoya (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-04-30
publication date: 2026-04-30
starting page: 1
ending page: 2
doi: 10.54105/ijam.A1228.06010426

Properties of Objects and Their Transformation on the Plane
type: item
description: Every function has its properties relative to the mechanisms of a circle, since cycles repeat. Such that all equations factor so well within its definition, that clearly, every circle has a radius that is greater than zero. If the radius expands by any set of numbers, then it is transforming. Hence, every object under it expands at a specific ratio. To a point that it is pretty natural for phenomena to repeat, given that it is within the area of its circumference. Although the events are not the same. Functions tend to approximate mathematical constants, primarily due to their periodicity of behaviour, as they are cyclic under certain conditions. The centre of a black hole is a point, so much like that of a circle. Since, therefore, objects alter the state of dimensions they occupy, so that relative to the point of reference, they either appear as paraboloids, ellipses, hyperbolas, or circles, depending on the context.
subject: Articles
  topic: section
subject: Transformation
  topic: keywords
subject: Objects
  topic: keywords
subject: Hyperbolas
  topic: keywords
creator: Avelanda
rights: Copyright (c) 2026 Avelanda (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-04-24
publication date: 2026-04-24
starting page: 19
ending page: 21
doi: 10.54105/ijam.A1166.04010424/

An Elementary Procedure in the Proof of Fermat's Last Theorem
type: item
description: Pierre de Fermat first stated, around 1637, that for any integer n > 2, the equation an + bn = cn has no positive integer solutions, and he wrote the theorem in the margin of a copy of Arithmetica. His proof is available only for the equation a 4 + b 4 = c 4 for the exponent n = 4. Subsequently, Euler proved the theorem in the equation a 3 + b 3 = c 3 for the exponent n = 3. Taking the above two proofs of Fermat and Euler, it would suffice to prove the theorem for n = p, where p is any prime > 3. In this proof, we hypothesize all r, s and t as positive integerssatisfying the equation rp + sp = tp and establish a contradiction. We use another auxiliary equation, x 3 + y 3 = z 3 , and combine the two equations using transformation equations. Solving the transformation equations, we establish a contradiction, thereby proving the theorem.
subject: Articles
  topic: section
subject: Transformation Equations
  topic: keywords
creator: P.N. Seetharaman
rights: Copyright (c) 2025 P.N. Seetharaman (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 95
ending page: 100
doi: 10.54105/ijam.A1227.05021025

Re-Evaluating the Value of π and Emerging New Concepts for Measuring Areas and Volumes Based on Translations and Rotations of Straight Lines, Planes, and Circles
type: item
description: The universal constant pie has been assigned a nonconverging value of approximately 3.145..., and this non-repeating, non-terminating decimal extends to millions—even trillions—of digits beyond the decimal point, as calculated by the most advanced computational methods. However, for thousands of years, mathematicians across different civilizations have attempted to determine the value of this fundamental constant in various ways. In geometry, angles are commonly expressed in terms of pie radians, and pie appears prominently in the formulas used to calculate the areas and volumes of curved geometrical figures such as circles, ellipses, spheres, cones, pyramids, and cylinders. Thus, pie has become an integral part of how we measure areas and volumes. Ancient scientists, in their efforts to understand and quantify the dimensions of curved or non-linear topological objects, sensed a recurring spatial constant underlying these forms. Over time, this invariant quantity came to be known as “pi.” The pursuit of its exact value has become a continuous endeavour spanning millennia, with each generation refining its estimation using increasingly sophisticated tools and methods. In this article, it is first emphasised that, although the mathematical parameters of area and volume have traditionally been expressed through formulas involving length, breadth, and height, their more profound physical significance has not been adequately explored. This work attempts to redefine area and volume fundamentally in terms of distance or length alone, offering new mathematical formulations for the areas and volumes of basic geometric shapes such as squares, cubes, circles, spheres, and others. Another critical shortcoming of conventional mathematics lies in its neglect of the reciprocal or inverse space of the universe when evaluating the value of pie. In recent research, it has been proposed that pie itself represents a circle whose radius corresponds to the smallest conceivable length in …
subject: Articles
  topic: section
subject: Pi (π)
  topic: keywords
subject: Reciprocal Geometry
  topic: keywords
subject: Area and Volume Redefinition
  topic: keywords
subject: Geometric Constants
  topic: keywords
subject: Cosmological Geometry
  topic: keywords
subject: non-Converging
  topic: keywords
subject: Planck Length
  topic: keywords
creator: Nishant Sahdev
creator: Chinmoy Bhattacharya
rights: Copyright (c) 2025 Nishant Sahdev, Chinmoy Bhattacharya (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 81
ending page: 94
doi: 10.54105/ijam.A1225.05021025

Hybrid Mean Value of General Quartic Gauss Sums and Three-Term Exponential Sums
type: item
description: This paper explores the hybrid power mean of three-term exponential sums, incorporating weights derived from general quartic Gauss sums. By employing the theory of Dirichlet characters in conjunction with fundamental properties of classical Gauss sums, we establish several significant results. Furthermore, we determine the corresponding weight function for these three-term exponential sums, with particular applications in coding theory.
subject: Articles
  topic: section
subject: Dirichlet Character
  topic: keywords
subject: Quartic Gauss Sums
  topic: keywords
subject: Three-Term Exponential Sum
  topic: keywords
creator: Dr. Shikha Singh
creator: Prof. Jagmohan Tanti
rights: Copyright (c) 2025 Dr. Shikha Singh, Prof. Jagmohan Tanti (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 74
ending page: 80
doi: 10.54105/ijam.B1219.05021025

An Index-Jets Framework for Irrationality: from Sturm-Liouville Operators to the AGM
type: item
description: We describe a general method that proves irrationality statements from second-order equations and from the arithmetic geometric mean (AGM). Let L Greater than sign 0 and define the bump Bn,L (x)=Qnn[x(L-x)]n (0 less than or equal to x less than or equal to L), where L=P/Q element of Q is in lowest terms. If u solves a second-order equation on [0,L] and its Prufer phase turns by an integer multiple of pie, then repeated integration by parts shows that In:=integral L0 Bn,L(x)dx is an integer combination of endpoint jets and of a fixed index term. Hence In element of Z (or Dn In element of Z for a controlled denominator Dn that depends only on finitely many endpoint Taylor coefficients of the coefficients of the equation). A Beta-function estimate gives 0 Less-than sign In less than or equal to L2n+1 Qnn/(2n+1) Long Rightwards Arrow n Long Rightwards Arrow the lemniscate 0 so for large n we have 0 Less-than sign In Less-than sign 1, which contradicts integrality. This scheme yields: (i) the irrationality of pie
subject: Articles
  topic: section
subject: Irrationality Statements
  topic: keywords
subject: Coefficients
  topic: keywords
subject: Function Estimate
  topic: keywords
creator: K. Srinivasa Raghava
creator: Dr. R. Sivaraman
rights: Copyright (c) 2025 K. Srinivasa Raghava, Dr. R. Sivaraman (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 70
ending page: 73
doi: 10.54105/ijam.B1224.05021025

Numerical and Analytical Method based Mathematical Approach to Estimate Water Pollution over 3-D Region
type: item
description: One of the major environmental issues is water pollution, which harms aquatic ecosystems, human health, and the sustainability of natural resources. To solve this problem, reliable mathematical models are required that can predict the behaviour of pollutants in three-dimensional (3-D) water systems. In this study, the temporal and spatial fluctuations of pollutant concentration in water are estimated using a 3-D diffusion model. The work's goals are to investigate the gradual rise in pollution levels in a 3-D region and assess the accuracy of analytical and numerical methods for addressing diffusion-based pollution issues. Two distinct mathematical techniques are employed to determine the level of pollution in water at a 3-D location: the Crank-Nicolson (C-N) method, a numerical method, and the Adomian Decomposition Method (ADM), an analytical method. The data collected from an experiment conducted in a 3-D cuboid tank, where water serves as the medium and iodised saltwater solution as the pollutant, are used to derive the initial and boundary conditions for the presented mathematical approaches. The dispersion of pollutants over time and space can be directly observed in this experiment, yielding crucial data for validating the mathematical model. The results of the experiments show that during all the time intervals, there is a rise in water pollution at all 3-D locations. Furthermore, the insignificant error in the form of parts per million (PPM) difference obtained when comparing the outcomes of the C-N with Experimental data (Exp. data) and ADM methodologies demonstrates the effectiveness of the proposed mathematical model. The benefit of this research lies in the use of mathematical approaches and experimental data to investigate water pollution in a 3-D region. The study demonstrates the validity of the proposed model and its ability to forecast the spread of pollution in water accurately. This method is also applicable to larger water systems than the experi…
subject: Articles
  topic: section
subject: Water Pollution
  topic: keywords
subject: 3-D Diffusion Equation
  topic: keywords
subject: Crank-Nicolson Method
  topic: keywords
subject: Adomian Decomposition Method Mathematics Subject Classification
  topic: keywords
subject: 35K57
  topic: keywords
subject: 65N06
  topic: keywords
creator: Tarjani Naik
creator: Dr. Mukesh Patel
creator: Dr. Rachna Patel
rights: Copyright (c) 2025 Tarjani Naik, Dr. Mukesh Patel, Dr. Rachna Patel (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 50
ending page: 62
doi: 10.54105/ijam.B1222.05021025

Complex Dynamics of an Aquatic Tri-Trophic Food Chain System with Holling Type IV Functional Response
type: item
description: Complex dynamics of a modified Hastings-Powell (HP) model (phytoplankton-zooplankton-fish) with Holling type IV functional response are investigated in this article. Boundedness of the system has been established. A detailed study of the boundary equilibrium points and their local stability has been carried out. The condition of uniform persistence of the system has also been derived. Dynamical complexities and subsequent changes in the states of the system have been portrayed using numerical simulation. Modified HP model with Holling type IV functional response gives rise to a similar type of chaotic dynamics (inverted 'teacup attractor’) as observed in the original HP model with Holling type II functional response. Chaotic or stable dynamics are also numerically verified using Sil'nikov eigenvalue analysis.
subject: Articles
  topic: section
subject: Holling type IV
  topic: keywords
subject: Boundedness
  topic: keywords
subject: Uniform persistence
  topic: keywords
subject: Chaos
  topic: keywords
creator: Dr. Partha Ghosh
rights: Copyright (c) 2025 Dr. Partha Ghosh (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 43
ending page: 49
doi: 10.54105/ijam.B1218.05021025

Comment on ‘The Triple Product Rule and the Subtleties of the Mathematical Tools Used in Thermodynamics’ by Felipe Américo Reyes Navarro and Rafael Edgardo Carlos Reye
type: item
description: It is shown that much of the criticism of my paper ‘The Triple Product Rule is Incorrect’ Indian Journal of Advanced Mathematics (IJAM), Vol. 1, No. 2, 2021. 1 − 3, https://doi.org/10.35940/ijam.B1102.101221 presented in [1] is incorrect.
subject: Articles
  topic: section
subject: Triple Product Rule
  topic: keywords
subject: Natural Variables
  topic: keywords
subject: Work
  topic: keywords
subject: Compression
  topic: keywords
subject: Function of Stat
  topic: keywords
creator: I.A. Stepanov
rights: Copyright (c) 2025 I.A. Stepanov (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 41
ending page: 42
doi: 10.54105/ijam.B1223.05021025

A Comparative Study on the Effects of Vertical and Horizontal Magnetic Fields on Double Diffusive Convection with Soret Effect in Couple-Stressed Nanofluids
type: item
description: This study aims to conduct a comparative analysis of double diffusive convection in a Soret-induced couple-stressed nanofluid layer under the influence of uniform vertical and horizontal magnetic fields. The linear stability analysis is based on the standard mode technique. The Galerkin method has been applied to find the critical Rayleigh number and the corresponding wave number in terms of various parameters. The effects of the Soret parameter, magnetic field, Lewis number, modified diffusivity ratio, Concentration Rayleigh-Darcy number, and Solutal Rayleigh number on the system's stability have been investigated. It has been observed that Stationary convection remains unaffected by the relaxation parameter, and the critical wave number is a function of both the couple stress parameter and the magnetic field; however, stationary convection is stabilised by the couple stress. In double diffusive convection under a magnetic field, the Darcy number also comes into play and has been observed to provide a stabilising effect on stationary convection. In comparison to ordinary fluids, convection sets up earlier in nanofluids. A comparative graphical analysis has been conducted to depict the effects of couple stress and Soret parameter on the stability of a nanofluid in the presence of both vertical and horizontal magnetic fields. It has been observed that the magnetic field still has a stabilising effect in both cases. However, in the case of a vertical magnetic field, the critical value of the stationary Rayleigh number is obtained at a significantly larger wave number compared to that in the case of a horizontal magnetic field, with all parameters remaining the same. The literature survey indicates that no study has investigated the comparison of the effect of vertical and horizontal magnetic fields on double diffusive convection in a couple stress nanofluid layer with a Soret factor. The present study examines the comparative analysis of the effect of vertical and hor…
subject: Articles
  topic: section
subject: Magnetic Field
  topic: keywords
subject: Rayleigh Number
  topic: keywords
subject: Couple-Stressed Nanofluid
  topic: keywords
subject: Soret Factor
  topic: keywords
creator: Suman Sindhwani
rights: Copyright (c) 2025 Suman Sindhwani (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 36
ending page: 40
doi: 10.54105/ijam.B1215.05021025

Completed Functor S-1() of the Localization Functor S-1(), Isomorphism and Adjunction
type: item
description: This article serves as a continuation of our previous work 1, which remains our primary reference for investigating specific homological properties with completion. Let the rings not be necessarily commutative and the modules be the unitary left (resp. right) modules. Let (G, (Gn) N element of N) be a filtered normal group equipped with the group topology associated with the filtration (Gn)nEN formed of normal subgroups and C(G) the set of Cauchy sequences with values in G. We define an equivalence relation R on C(G) by: (xn)R(yn) (xn)-(yn)= (xn-yn) converges to 0, noted by (xn-yn) 0. The quotient set C(G)/R:+{(xn)| (xn)element of C(G)} denoted G is equipped with a group structure and is called the completed groupe of G. For any filtered ring (resp. left A-module) (A, (In)nelement ofN (resp.(M,(Mn)nelement of N), the completed group A (resp. M) is equipped with a ring structure (resp. A-module) by (an ) x (bn ) = (anbn ) (resp.(an).(mn)=(an.mn)) where (an), (bn) element of A(resp. (mn ) element of M) called completed ring (resp. module) of A (resp. M). And for all saturated multiplicative subset S of A that satisfies the left Ore conditions, S = {(xn) element of A ∣ (xn ) not-equal 0 and ∃ n0 element of N, n greater than or equal to n0,xn element of S} is a saturated multiplicative subset of A that satisfies the left Ore conditions 1. Among the main results of this article, we have : – the functors S-1 () is isomorphic toS-1(A) tensor product A−. and S-1 () is isomorphic to S-1(A) tensor productA−. – the functors Home A(S-1A tensor productAM,−) and HOM A(S-1Atensor productA M,−) are isomorphic. – the functors S-1Atensor productA – and HomA(S-1A,−) are adjoints. This Study Allows How Establish a Relationship Between Completion [2] and Localization [4] Under the Assumptions of a Topological Structure.
subject: Articles
  topic: section
subject: Ring
  topic: keywords
subject: Modules
  topic: keywords
subject: Filtration
  topic: keywords
subject: Completion
  topic: keywords
subject: Ore Condition
  topic: keywords
subject: Localization
  topic: keywords
subject: Isomorphisms
  topic: keywords
subject: Categories
  topic: keywords
subject: Functors
  topic: keywords
subject: Completed Functor
  topic: keywords
subject: Adjunction
  topic: keywords
creator: Abdoulaye Mane
creator: Mohamed Ben Maaouia
creator: Mamadou Sanghare
rights: Copyright (c) 2025 Abdoulaye Mane, Mohamed Ben Maaouia, Mamadou Sanghare (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 27
ending page: 35
doi: 10.54105/ijam.B1214.05021025

Study of Some Degree Five Identities of Type (4,1)
type: item
description: In non-associative algebra, irreducible identities of degree five are the least studied. Following Osborn's studies, the only identity of type (5) has generated very little literature, as seen in ([1]) and ([2]). Hence, our interest in identities of the following type. The purpose of this study is to enable us to consider a baric case study at a later stage, such as in ([3]) and ([4]). This paper is devoted to the study of three of type (4,1), taken from the families of irreducible degree five identities of Osborn. We conduct this study in the presence of an idempotent, through a Peirce decomposition, depending on whether the Peirce polynomial is reducible or not over the base field of the algebra. In each studied case, we find two orthogonal subalgebras. Therefore, in the first two studied identities, we manage to show that there is a homomorphism over one of the subspaces of A, whose kernel is an ideal.
subject: Articles
  topic: section
subject: Idempotent
  topic: keywords
subject: Linearization
  topic: keywords
subject: Peirce Decomposition
  topic: keywords
creator: Dr. Abdoulaye DEMBEGA
rights: Copyright (c) 2025 Dr. Abdoulaye DEMBEGA (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 18
ending page: 26
doi: 10.54105/ijam.B1212.05021025

Existence and Uniqueness of Weak Solutions to the Fractional Navier-Stokes Equations in Turbulent Flows: A Study of Boundedness and Stability
type: item
description: This study employs fractional analysis to investigate the existence and uniqueness of weak solutions to the fractional NavierStokes equations in three-dimensional turbulent flows, thereby addressing the complexity inherent in turbulent regimes. Turbulence presents significant challenges in fluid dynamics, characterised by chaotic and erratic motion that confounds conventional modelling techniques. By reformulating the NavierStokes equations using fractional derivatives, we capture non-local effects and memory phenomena, thereby enhancing the mathematical representation of fluid behaviour. First, we transform the Navier-Stokes equations to demonstrate the utility of fractional analysis. The second thing we do is show that there are weak answers in some situations. This means that we can use our models with starting data that isn't stable. This is the third thing we do. We show that these answers are unique. This proves that our models are right. Ultimately, we demonstrate that weak solutions remain bounded over time under specific conditions regarding the initial data and external forces. When the Reynolds number approaches a critical level, we investigate its stability. This helps us understand how smooth flow can become rough flow. The findings of this research not only advance the theoretical understanding of weak solutions to the fractional Navier-Stokes equations but also have practical implications for modeling complex fluid systems. By linking fractional derivatives with turbulent flows, this work contributes to the broader field of fluid dynamics, paving the way for future investigations in applied mathematics and engineering. Ultimately, this exploration enhances our understanding of turbulence and its mathematical foundations, emphasising the importance of fractional calculus in accurately modelling fluid dynamics.
subject: Articles
  topic: section
subject: Navier-Stokes Equations
  topic: keywords
subject: Weak Solutions
  topic: keywords
subject: Turbulent Flows
  topic: keywords
subject: Existence and Uniqueness
  topic: keywords
subject: Boundedness
  topic: keywords
subject: Stability Analysis
  topic: keywords
creator: Dr. Raghad Sahib Shamsah
rights: Copyright (c) 2025 Dr. Raghad Sahib Shamsah (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 8
ending page: 17
doi: 10.54105/ijam.B1211.05021025

An Innovative Approach to Find Remainder
type: item
description: Let r be the remainder when x! is divided by p, (x, p) ∈ N, x < p. Then the value of r is given by the minimum value of k for which (−1) (m-1) (m-1)! k + 1 = 0(mod p), where m = p − x, k ∈ N.
subject: Articles
  topic: section
subject: Remainder
  topic: keywords
subject: Number Theorem
  topic: keywords
subject: Factorial Remainder
  topic: keywords
subject: Remainder using Wilson’s Theorem
  topic: keywords
creator: Prof. Anil Kumar Sharma
rights: Copyright (c) 2025 Prof. Anil Kumar Sharma (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 6
ending page: 7
doi: 10.54105/ijam.B1205.05021025

Some Results on Specificity of Possibility Distributions
type: item
description: Specificity of a possibility distribution is akin to the entropy of a probability distribution. It serves an essential purpose to zero in on the maximum probability observation. However, when we discuss the existing definition of possibility distribution, it lacks applicability in real-world problems; hence, specificity also becomes an underrated measure for gauging the degree of uncertainty in a possibility distribution. In this paper, we present new findings on the specificity of a possibility distribution, resulting from our research on data-based semantic information analysis in hybrid human-machine systems. In this research, we propose a new frequency-based possibility and probability measure and formalise a new method for fitting restrictions on data or information available in the system. We will demonstrate that the proposed formula is superior to existing specificity measures and discuss various applications of specificity measures in solving problems related to hybrid systems. We shall summarise this paper by providing a real-world application of the proposed measure.
subject: Articles
  topic: section
subject: Possibility Distribution
  topic: keywords
subject: Restriction
  topic: keywords
subject: Specificity
  topic: keywords
subject: Hybrid System
  topic: keywords
subject: Semantic Information
  topic: keywords
creator: Dr. Jayesh V. Karanjgaonkar
rights: Copyright (c) 2025 Dr. Jayesh V. Karanjgaonkar (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 1
ending page: 5
doi: 10.54105/ijam.B1206.05021025