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An Analytic Proof of Bugeaud-Mignotte Theorem

Jamel Ghanouchi
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.
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  How to cite this article:

Jamel Ghanouchi , 2009. An Analytic Proof of Bugeaud-Mignotte Theorem. Asian Journal of Mathematics & Statistics, 2: 9-14.

DOI: 10.3923/ajms.2009.9.14



11, 111, 1111 in base ten are not pure powers, we can easily verify it, but is it true for a number of n digits equal 1 in base ten, for all n. This study is an answer to this question. A number with n digits equal to 1 (111…1) in base ten is equal to a number with n digits equal to 9 (999…9) in base ten divided by 9, which means , the theorem stipulates that , . This research proposes an original and analytic proof of this theorem called Bugeaud and Mignotte theorem (Bugeaud and Mignotte, 1999).


Let Bugeaud-Mignotte equation:

Then with j2 = -1
10n-9Yq = 1 = 10n+j(9jYq)
We pose:
10n = x, 9jYq = y, u = 1, z = 9jYq10n

Lemma 1



Proof of Lemma 1

We pose:
until infinty.

Lemma 2

The expression of the sequences is

Proof of Lemma 2

For I = 2, , it is verified. We suppose the expressions true for I

Lemma 3

xi-jyi = x-jy

Lemma 4

The lemmas 1, 2, 3 imply that jy-x = 0

Proof of Lemma 4

We pose:
We pose:
If we make the hypothesis that
The hypothesis is false and
In all cases
is convergent, then the general term of the series tends to zero
And it is impossible: the existence of Y and q is impossible!


Bugeaud-Mignotte equation has effectively no solution, and an analytic proof exists. The generalization is is it possible? It seems that there are only three solutions:

Bugeaud, Y. and M. Mignotte, 1999. Sur l'équation diophantienne xn-/x-1x-1 = yq, II = On the diophantine equation xn-1-x-1 = yq, II. Comptes Rendus de l'Académie des Sciences. Série 1, Mathématique, 328: 741-744.
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