Volume-4 Issue-1, April 2024 – Indian Journal of Advanced Mathematics (IJAM)

A119005010425

IJAM Journal — Thu, 13 Feb 2025 07:35:50 +0000

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. The journal aims to publish high-quality peer–reviewed original articles in the area of Mathematics that covers Applied Mathematics, Arithmetic, Algebra, Calculus and Analysis, Geometry and Topology, Trigonometry, Combinatorics, Logic, Number Theory, Numerical Analysis, Dynamical Systems, Differential Equations, Mathematical Physics, Information Theory and Signal Processing, Probability and Statistics, Game Theory, Operations Research, Discrete Mathematics, Business Mathematics and Financial Mathematics. #Applied Mathematics #Arithmetic #Algebra #Calculus and Analysis #Geometry and Topology #Trigonometry #Combinatorics #Logic #Number Theory #Numerical Analysis #Dynamical Systems #Differential Equations #Mathematical Physics #Information Theory and Signal Processing #Probability and Statistics #Game Theory #Operations Research #Discrete Mathematics #Business Mathematics and Financial Mathematics #PhD ademic #Scopus #SCI #LatticeScience #Springer, #ScienceDirect #IEEE #Mendeley #Research #Scholarship #UGC #SSRN #LatticeScience #ESCI #Science #Journal #Conference #SSRN #PubLons

In Search of an Elementary Proof for Fermat’s Last Theorem
P. N. Seetharaman

P. N Seetharaman, (Retired Executive Engineer, Energy Conservation Cell), Tamil Nadu State Electricity Board, Tamil Nadu, India.

Open Access | Editorial and Publishing Policies | Cite | Zenodo | OJS | Indexing and Abstracting
© The Authors. Published by Lattice Science Publication (LSP). This is an open-access article under the CC-BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Abstract: 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 .

Keywords: Transformation Equations To Two Fermat’s Equations, Mathematics Subject Classification 2010: 11A–XX.
Scope of the Article: Applied Mathematics

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A118105010425

IJAM Journal — Fri, 27 Dec 2024 09:37:13 +0000

A Comprehensible Proof for Fermat’s Last Theorem
P. N. Seetharaman

P. N Seetharaman, (Retired Executive Engineer, Energy Conservation Cell), Tamil Nadu State Electricity Board, Tamil Nadu, India.

Abstract: 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.

Keywords: Transformed Fermat’s Equations through Parameters. 2010 Mathematics Subject Classification 2010: 11A–XX.
Scope of the Article: Applied Mathematics

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B117404021024

IJAM Journal — Thu, 25 Jul 2024 10:23:57 +0000

Solution of Brocard’s Problem
M.I. Karimullah

Mohamed Imteaz Karimullah, Burlington, Ontario, Canada.

Abstract: 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.

Keywords: Brocard’s Problem, Diophantine Equation, Brown Numbers.
Scope of the Article: Discrete Mathematics

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B117204021024

IJAM Journal — Fri, 28 Jun 2024 09:35:57 +0000

On Approach for Analysis of Distribution of Temperature During Effects of Radiation on Biological Tissues
E. L. Pankratov

Evgeny Leonidovich Pankratov, Department of Mathematical and Natural Sciences, Nizhny Novgorod State University, Nizhny Novgorod, Nizhny Novgorod region, Russia.

Abstract: Different types of irradiations play an essential role in biological processes directly related to human health. They are currently widely used in clinical practice. In this situation, several studies have been conducted to determine the therapeutic effects achieved through the actions of irradiation on living organisms. One type of irradiation of tissues of the considered organisms is electromagnetic radiation. A consequence of radiation exposure in organisms is the heating of their cells. In this paper, we introduce a model and an analytical approach to analyse the distribution of temperature because of radiation on biological tissues. We consider the possibility of controlling heat transport in biological tissues by adjusting the heating conditions. Additionally, we present an analytical approach for analysing the obtained model. The approach considers the changing parameters of the thought processes on both coordinates and time, as well as their nonlinearity.

Keywords: Effect of Radiation; Biological Tissues; Prognosis of Heat Transport; Control of Heat Transport; Analytical Approach for Analysis.
Scope of the Article: Discrete Mathematics

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A116604010424

IJAM Journal — Tue, 30 Apr 2024 11:24:40 +0000

Properties of Objects and Their Transformation on the Plane
Avelanda

Avelanda, Department of Mathematics, University of the Free State, Summerstrand, South Africa.

Abstract: 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.

Keywords: Transformation, Objects, Hyperbolas
Scope of the Article: Discrete Mathematics

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A116804010424

IJAM Journal — Tue, 30 Apr 2024 04:00:56 +0000

On Some Relations Involving the Ramanujan’s Tau Function
R. Sivaraman¹, J. López-Bonilla², S. Vidal Beltrán³

¹Dr. R. Sivaraman, Department of Mathematics, Dwaraka Doss Goverdhan Doss Vaishnav College, Chennai (Tamil Nadu), India.

²Prof. J. López-Bonilla, ESIME-Zacatenco, Instituto Politécnico Nacional, Edif. 4, 1er. Piso, Col. Lindavista CP 0778, CDMX, México.

³S. Vidal Beltrán, ESIME-Zacatenco, Instituto Politécnico Nacional, Edif. 4, 1er. Piso, Col. Lindavista CP 0778, CDMX, México.

Abstract: a recurrence relation is known for Ramanujan’s tau-function involving the sum of divisors function𝝈(𝒏), whose solution gives a closed formula for 𝝉(𝒏) in terms of complete Bell polynomials, and a determinantal expression for 𝝈(𝒎) where the values participate 𝝉(𝒌).

Keywords: Sum of divisors function, Recurrence relations, Ramanujan’s function 𝝉(𝒏), Bell polynomials, Chebyshev polynomials of the second kind.
Scope of the Article: Discrete Mathematics

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A116704010424

IJAM Journal — Tue, 30 Apr 2024 03:53:48 +0000

Some Algorithms of Graph Theory in Cryptology
Auparajita Krishnaa

Auparajita Krishnaa, Department of Mathematics and Statistics, Mohan Lal Sukhadia University, Udaipur (Rajasthan) India.

Abstract: The inventive use of concepts from Graph Theory plays a significant role in hiding the original plaintext, resulting in a significantly safer data transfer. In this work, tree traversal algorithms such as Inorder, Preorder, and Postorder, as well as Kruskal’s algorithm for creating a minimal spanning tree, and the modified graph labelling scheme of graceful labelling, which allows for the repetition of exactly one vertex label for certain graphs, have been employed to create highly hidden Cypher Texts. Encryption and decryption algorithms for all these methods are being presented in this work.

Keywords: Modified Graceful Labelling, Tree Traversal Algorithms, Expression Tree, Modified Ordered Rooted Tree, Basic-Cipher-text, Cipher-Matrix (of different kinds), Encryption-Array.
Scope of the Article: Discrete Mathematics

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A116504010424

IJAM Journal — Tue, 30 Apr 2024 03:46:34 +0000

On Solving a Quadratic Diophantine Equation Involving Odd Powers of 17
J. López-Bonilla¹, R. Sivaraman²

¹Prof. J. López-Bonilla, ESIME-Zacatenco, Instituto Politécnico Nacional, Edif. 4, 1er. Piso, Col. Lindavista CP 0778, CDMX, México.

²Dr. R. Sivaraman, Associate Professor, Department of Mathematics, Dwaraka Doss Goverdhan Doss Vaishnav College, Chennai (Tamil Nadu), India.

Open Access | Editorial and Publishing Policies | Zenodo | OJS | Indexing and Abstracting
© The Authors. Published by Lattice Science Publication (LSP). This is an open-access article under the CC-BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Abstract: Diophantine Equations, named after the ancient Greek mathematician Diophantus, play a vital role not only in number theory but also in several branches of science. In this paper, we will solve one of the quadratic Diophantine equations where the right-hand side is an odd positive integral power of 17 and provide its complete solutions. The method adopted to solve the given equation is based on the concept of the polar form of a particular complex number. This concept can be generalised to solve similar equations.

Keywords: Quadratic Diophantine Equation, Polar Form, Euler’s Formula, Positive Integer Solutions.
Scope of the Article: Number Theory

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A116404010424

IJAM Journal — Tue, 30 Apr 2024 03:37:43 +0000

The Triple Product Rule and the Subtleties of the Mathematical Tools Used in Thermodynamics
Felipe Américo Reyes Navarro¹, Rafael Edgardo Carlos Reyes²

¹Felipe Américo Reyes Navarro, Facultad de Ciencias Físicas, Universidad Nacional Mayor de San Marcos, Calle Germán Amézaga 375, Cercado, Lima, Perú.

²Rafael Edgardo Carlos Reyes, Facultad de Ciencias Físicas, Universidad Nacional Mayor de San Marcos, Calle Germán Amézaga 375, Cercado, Lima, Perú.

Abstract: We have clarified several formulas from mathematics and physics. Specifically, we have demonstrated that several standard concepts from thermodynamics, which have been recently investigated and rejected in some papers, remain valid. Thus, we have studied the triple product rule, the first law of thermodynamics, Mayer’s relation, work, the function of state, and the relationship between the coefficient of thermal expansion, the coefficient of isothermal compressibility, and the thermal coefficient of pressure. We have utilised multivariable calculus, properties of Jacobians, and other mathematical tools. Likewise, we have provided several didactic examples concerning the interpretation of thermodynamic and mathematical formulas. We have concluded that the basis of several thermodynamic formulas is correct, but it is also complete with subtleties. Thus, those formulas continue to be valid despite some claims in the recent literature.

Keywords: Triple Product Rule, Jacobians, Natural Variables, Work, Function of State.
Scope of the Article: Mathematical Physics

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