Solution of diophantine equation x⁴ + y⁴= pᵏz³ for primes 2 ≤ p ≤ 13. - Ismail, Shahrina (2011)
he purpose of this study is to determine the existence, types and the cardinality of the solutions for the diophantine equation x⁴ + y⁴=z³ and x⁴ + y⁴=pᵏz³ for p a prime, 2≤ p≤13 and k∈Z+in the rings of integers Z and Gaussian integers Z(i). Another aim of this study was to develop methods of finding all solutions to these equations. In finding solutions for the diophantine equation x⁴ + y⁴=pᵏz³ in the rings of integers and Gaussian integers, the values of (p,k) are restricted to (p,k) = {(1,1),(2,1),2,k),(3,k).(5,k),(7,k),(11,k),(13,k}. Our research begins by determining the patterns of solutions to these equations. Based on our observation on these patterns, we determine the general form of solution to the equations. Tools and methods in number theory such as divisibility, congruences, properties of prime numbers and method of proof by contradiction are applied in solving these types of iv diophantine equations. Our result shows that there exist infinitely many solutions to these types of diophantine equations in both rings of integers and Gaussian integers for both cases x=y and x≠y. The main result obtained is formulation of a generalized method to find all the solutions for both types of diophantine equations.