# Solutions to Some Advanced Methods in Solving Diophantine Equations

Chapter

## Abstract

1.Solve the equation
$$x^2 + 4 = y^n,$$
where n is an integer greater than 1. Solution. For n = 2, the only solutions are (0, 2) and (0, –2). For n = 3, we have seen in Example 4 that the solutions are (2, 2), (–2, 2), (11, 5), and (–11,5). Lef now n ≥ 4. Clearly, for n even, the equation is not solvable, since no other squares differ by 4. For n odd, we may assume without loss of generality that n is a prime p ≥ 5. Indeed, if n = q k , where q is an odd prime, we obtain an equation of the same type: x2 + 4 = (y k ) q .

## Keywords

Positive Integer Fundamental Solution Prime Divisor Advance Method Diophantine Equation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Authors and Affiliations

• Titu Andreescu
• 1
Email author
• Dorin Andrica
• 2
• 3
• Ion Cucurezeanu
• 4
1. 1.School of Natural Sciences and MathematicsUniversity of Texas at DallasRichardsonUSA
2. 2.Faculty of Mathematics and Computer ScienceBabeş-Bolyai UniversityCluj-NapocaRomania
3. 3.Department of Mathematics College of ScienceKing Saud UniversityRiyadhSaudi Arabia
4. 4.Faculty of Mathematics and Computer ScienceOvidius University of ConstantaConstantaRomania