On the diophantine equation Px + Qy = z² - Japar, Izzati Izyani; Sapar, Siti Hasana; M. Johari, M. Aidil (Faculty of Science and Natural Resources, Universiti Malaysia Sabah, 2024)
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.