An Analytic Proof of Bugeaud-Mignotte Theorem

The Bugeaud-Mignotte theorem concerns the Diophantine equation , when X = 10. It includes the fact that any integer greater than 2 and with all digits equal to 1 in base ten cannot be a pure power. It means that does not exist Y an integer strictly greater than 1 and q an integer strictly greater than 1 for which Yq is a number with all digits equal to 1 in base ten.