Abstract
In the present article, we extend previous results of the author and we show that when K is any quadratic imaginary field of class number one, Fermat's equation ap+bp+cp=0 does not have integral coprime solutions a,b,c∈K∖{0} such that 2|abc and p≥19 is prime. The results are conjectural upon the veracity of a natural generalisation of Serre's modularity conjecture.
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