Application of fields formed by the Gauss periods to the investigation of cyclic diophantine equations

A. V. Tolstikov

Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 4, Tome 67 (1977), pp. 201-222

Résumé

The question of the nonsolvability of the equation $$ Z^*_l(x_0,x_1,\dots,x_t)=\prod^{l-1}_{i=0}\sum^t_{j=0}x_j\zeta^{ij}=Dl^wx^l,\quad (D\varphi(D)z,l)=1 $$ in rational integers $x_0,x_1,\dots,x_t,z$ satisfying certain additional conditions is investigated. Two cases are considered: 1) $l$ is a regular prime number and $0<t<l-1$; 2) $l$ is an irregular prime number, $l=fe+1$ ($f$ is prime), $l>c_0(f,t)$ and $l$ does not divide the Bernoulli numbers $B_{fk+1}$ ($k=1,3,\dots,e-1$), $B_{2fk}$ ($k=1,2,\dots,\frac{e}{2}-1$).