On a Class of Solutions for the Hyperbolic Diophantine Equation

Kannan, J.; Somanath, Manju; Raja, K.

doi:10.12732/ijam.v32i3.6

Cited by 4 publications

(4 citation statements)

References 3 publications

(2 reference statements)

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“…) for are found by Kannan et al [19]. It is also shown that the Diophantine equation , -, -,is solvable in prime variables , and with some conditions on [20].…”

Section: Introductionmentioning

confidence: 81%

Does the Solution to the Non-linear Diophantine Equation 3x+35y=Z2 Exist?

Biswas¹

2022

J. Sci. Res.

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This paper investigates the solutions (if any) of the Diophantine equation 3x + 35y = Z2, where , x, y, and z are whole numbers. Diophantine equations are drawing the attention of researchers in diversified fields over the years. These are equations that have more unknowns than a number of equations. Diophantine equations are found in cryptography, chemistry, trigonometry, astronomy, and abstract algebra. The absence of any generalized method by which each Diophantine equation can be solved is a challenge for researchers. In the present communication, it is found with the help of congruence theory and Catalan’s conjecture that the Diophantine equation 3x + 35y = Z2 has only two solutions of (x, y, z) as (1, 0, 2) and (0, 1, 6) in non-negative integers.

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“…) for are found by Kannan et al [19]. It is also shown that the Diophantine equation , -, -,is solvable in prime variables , and with some conditions on [20].…”

Section: Introductionmentioning

confidence: 81%

Does the Solution to the Non-linear Diophantine Equation 3x+35y=Z2 Exist?

Biswas¹

2022

J. Sci. Res.

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“…The two key functions of cryptography are encryption and decryption. Data properly encryption and decryption are very much necessary in communication systems [27].…”

Section: Proposed Workmentioning

confidence: 99%

Secure nano-communication framework using RSCV cryptographic circuit in IBM Q

Kundu,

Das,

Debnath

et al. 2023

Phys. Scr.

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In the cryptographic domain, quantum and its real-time hardware simulation make it easier to secure data during communication. Here, using quantum logic, a unique encryption technique called Reversible select, cross, and variation (RSCV) encryption and decryption, which involves swapping input data halves, is shown. In this article using IBM Q, we created a cryptographic encoder and decoder circuit design utilizing various quantum gates. Based on the encoder/decoder circuit, a simple nanocommunication framework is proposed. Further, to explore the application of the noise model, how to utilize this model to create noisy replicas of these quantum circuits to research the impacts of noise that occur for actual device output is shown. To reduce measurement mistakes, measurement calibration is performed using qiskit ignis model. Preparing all 2n basis input states and calculating the likelihood of counting in the other basis states are the key concepts. The percentage improvement we achieved is 40%, 30%, and 30%, respectively, compared to earlier ones, in RSCV encryption, decryption, and RSCV cryptographic communication architecture for fake provider noise error model. It is feasible to adjust the average outcomes of an additional interesting experiment using these calibrations.

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“…There are a few quartic Diophantine equations that must be solved to arrive at the non-existence. To resolve those equations, the method of variable transformation is adopted as in (6,7) . Also, the basic number theoretic concepts are retrieved from "Fundamental Perceptions in Contemporary Number Theory" (8) , contributed by J.…”

Section: Introductionmentioning

confidence: 99%

2-Peble Triangles Over Figurate Numbers

Mahalakshmi,

Kannan,

Deepshika

et al. 2023

IJST

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Objectives:The main aim of this article is to discuss the existence or nonexistence of 2-Peble triangles over some figurate numbers. Methods: A few quartic equations over integers are solved to complete the objective at hand. This is done with the aid of the transformation of variables. Additionally, fundamental concepts such as mathematical induction and parity of integers are used. Findings: Here, it is demonstrated that there are no 2-Peble triangles over triangular, hexagonal, and octagonal numbers. The same process is explained for particular special numbers as an exceptional instance. Novelty: This article defines a triangle, the d-Peble triangle over figurate numbers, which creates a link between the Pell equation and a common geometric shape. So many previous researchers, when examining a problem involving geometric shapes, attain their expected result using Diophantine equations. But this concept differs from those as this uses figurate numbers and a Pell equation to create a triangle.

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On a Class of Solutions for the Hyperbolic Diophantine Equation

Cited by 4 publications

References 3 publications

Does the Solution to the Non-linear Diophantine Equation 3<sup>x</sup>+35<sup>y</sup>=Z<sup>2</sup> Exist?

Does the Solution to the Non-linear Diophantine Equation 3<sup>x</sup>+35<sup>y</sup>=Z<sup>2</sup> Exist?

Secure nano-communication framework using RSCV cryptographic circuit in IBM Q

2-Peble Triangles Over Figurate Numbers

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