On The Group of Units Classification In 3-Cyclic and 4-cyclic Refined Rings of Integers And The Proof of Von Shtawzens' Conjectures

Abstract

First Von Shtawzen's Diophantine equation is a non-linear Diophantine equation with three variables . This equation has been conjectured that it has a finite number of integer solutions, and this number of solutions is divisible by 6. Second Von Shtawzen's Diophantine equation is a non-linear Diophantine equation with four variables. This equation has been conjectured that it has a finite number of integer solutions, and this number of solutions is divisible by 8. In this paper, we prove that first Von Shtawzen's conjecture is true, where we show that first Von Shtawzen's Diophantine equations has exactly 12 solutions. On the other hand, we find all solutions of this Diophantine equations. In addition, we provide a full proof of second Von Shtawzen's conjecture, where we prove that the previous Diophantine equation has exactly 16 solutions, and we determine all of its possible solutions