On the equation \(a^ 3+b^ 3+c^ 3=d^ 3\)
The author deals with the solution of diophantine equation \[ A^3+B^3+C^3=D^3.\leqno(1) \] The purpose of the paper is to solve (1) in quadratic forms of two variables and characterize the solutions obtained this way. The following results are proved (Theorems 1-4): If \((\omega_1,\omega_2,\omega_3,\omega_4)\) is a nontrivial (i.e. \(\omega_4\neq \omega_i\), \(i=1,2,3\)) integer solution of (1), then for any integer values of \(a\) and \(b\) the integers \((z_1,z_2,z_3,z_4)\) satisfy (1), where \(z_i=c_{i1}a^2+c_{i2}ab+c_{i3}b^2\), \(i=1,2,3,4\) and \(c_{ij}\) depends on \((\omega_1,\omega_2,\omega_3,\omega_4)\) in an explicit way. The solutions \((z_1,z_2,z_3,z_4)\) obtained from a given solution \((\omega_1,\omega_2,\omega_3,\omega_4)\) can be characterized by the condition \({\omega_1+\omega_3\over \omega_4-\omega_2}={z_1+z_3\over z_4-z_2}\). The set of these fractions \({\omega_1+\omega_3\over \omega_4-\omega_2}\) is also characterized. The Theorems 5-7 generalize Ramanujan's formula, which also gives a parametric solution of (1).
scientific article; zbMATH DE number 1099660