A REFINED MODULAR APPROACH TO THE DIOPHANTINE EQUATION x(2) + y(2n) = z(3)

作者：Dahmen Sander R^*

来源：International Journal of Number Theory, 2011, 7(5): 1303-1316.

DOI：10.1142/S1793042111004472

• Generalized Fermat equation
• modular form
• Galois representation
• elliptic curve

Let n be a positive integer and consider the Diophantine equation of generalized Fermat type x(2) + y(2n) = z(3) in nonzero coprime integer unknowns x, y, z. Using methods of modular forms and Galois representations for approaching Diophantine equations, we show that for n is an element of {5, 31} there are no solutions to this equation. Combining this with previously known results, this allows a complete description of all solutions to the Diophantine equation above for n <= 10(7). Finally, we show that there are also no solutions for n equivalent to -1 (mod 6).

• 出版日期2011-8