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On the diophantine equation Px + Qy = z²


Citation

Japar, Izzati Izyani and Sapar, Siti Hasana and M. Johari, M. Aidil (2024) On the diophantine equation Px + Qy = z². Borneo Science (The Journal of Science and Technology), 45 (2). pp. 1-10. ISSN 1394-4339; eISSN: 2231-9085

Abstract

Diophantine equation is a polynomial equation with two or more unknowns which integral solutions are required. An exponential Diophantine equation is an equation that has additional variable or variables occurring as exponents. This paper concentrates on finding an integral solution to the Diophantine equation px + qy = z2 for x + y = 5 and p, q are twin primes, cousin primes, sexy primes and also for any positive integers. By looking at the pattern of the solutions of each case, the theorems and lemmas will be constructed. In this paper, for all the possibilities of x + y = 5, it is proved that the Diophantine equation has no non-trivial solution for twin primes, cousin primes and sexy primes whereas for any positive integers, the equation has infinitely many solutions.


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Additional Metadata

Item Type: Article
Divisions: Faculty of Science
Institute for Mathematical Research
DOI Number: https://doi.org/10.51200/bsj.v45i2.5878
Publisher: Faculty of Science and Natural Resources, Universiti Malaysia Sabah
Keywords: Diophantine equation; Integral solution; Twin primes; Cousin primes; Sexy prime
Depositing User: Ms. Che Wa Zakaria
Date Deposited: 24 Sep 2025 03:25
Last Modified: 24 Sep 2025 03:25
Altmetrics: http://www.altmetric.com/details.php?domain=psasir.upm.edu.my&doi=10.51200/bsj.v45i2.5878
URI: http://psasir.upm.edu.my/id/eprint/120155
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