Solutions to Some Advanced Methods in Solving Diophantine Equations



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 .


Positive Integer Fundamental Solution Prime Divisor Advance Method Diophantine Equation 
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Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  • Titu Andreescu
    • 1
  • 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

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