ABSTRACT
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
DOI: 10.3923/ajms.2009.9.14
URL: https://scialert.net/abstract/?doi=ajms.2009.9.14
INTRODUCTION
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).
THE PROOF
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
(1) |
(2) |
Proof of Lemma 1
And: |
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 |
But: |
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: |
And: |
And: |
If we make the hypothesis that |
And: |
But: |
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! |
CONCLUSION
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:
REFERENCES
- 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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