Solutions to Some Advanced Methods in Solving Diophantine Equations
- 1.7k Downloads
1.Solve the equation
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 .
$$x^2 + 4 = y^n,$$
KeywordsPositive 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.
Unable to display preview. Download preview PDF.
© Birkhäuser Boston 2009