Patterns of Integer Solutions to Homogeneous Ternary Quadratic Equation ax²+by²=(a+b)(c²+ab)z²

Abstract This paper aims at determining many non-zero distinct integer solutions to the homogeneous ternary quadratic Diophantine equation ax²+by²=(a+b)(c²+ab)z², where a,b are non-zero distinct positive integers and the product a*b is square-free. Substitution technique and factorization method are utilized to obtain the patterns of solutions in integers for the given homogeneous ternary quadratic equation. Keywords: Homogeneous quadratic equation, Ternary quadratic equation, Integer solutions, Transformation technique, Factorization method