On the unit equation over cyclic number fields of prime degree

Let $\ell \ne 3$ be a prime. We show that there are only finitely many cyclic number fields $F$ of degree $\ell$ for which the unit equation $λ+ μ= 1, \qquad λ,~μ\in \mathcal{O}_F^\times$ has solutions. Our result is effective. For example, we deduce that the only cyclic quintic number field for which the unit equation has solutions is $\mathbb{Q}(ζ_{11})^+$.

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