The Diophantine Equation X³ = u+v over Real Quadratic Fields
Let k be a real quadratic field and let and be the ring of integers and the group of units, respectively. A method of solving the Diophantine equation X³ = u+v (, ) is developed.
Let k be a real quadratic field and let and be the ring of integers and the group of units, respectively. A method of solving the Diophantine equation X³ = u+v (, ) is developed.
We investigate the average order of the divisor function at values of binary cubic forms that are reducible over and discuss some applications.
We show that the Diophantine equation of the title has, for , no solution in coprime nonzero integers and . Our proof relies upon Frey curves and related results on the modularity of Galois representations.