The Legendre equation in Euclidean imaginary quadratic number fields

Abstract

We show that the equation x2+y2+z2=0, which has no nontrivial solutions in the ring of integers of Q(−7), does not have them in other quadratic number field extensions either.Then, we show that if the Legendre's equation with coefficients a,b,c in the ring of integers of Q(−d), for d=1,2,3,7,11, has a solution (x,y,z), it has a solution witha)|z0|≤43−d|ab|, for d=1,2b)|z0|≤16d−d2+14d−1|ab|, for d=3,7,11.This represents an improvement, in the case d=1, of bounds given previously.