ABSTRACT
(MSC = 11) more than one century ago, the Belgian mathematician Eugene Catalan has formulated a famous conjecture. It became a theorem in 2004. The theorem stipulates that the following equation Yp = 1+Xq has only one solution, which is 32 = 1+23 when X>1, Y>1, p>1, q>1 all integers. We prove in this research firstly that Catalan equation is equivalent to the following equation Yq-p = Xp-1. After a little change of the data of the problem, we prove also that Catalan equation implies two other equations. Those equations allow to define convergent sequences. It is the Algebraic-Analytic approach which conducts to the impossibility of Catalan equation for p>2. The equation is simplified to the case p = 2, q = 3. It becomes consequently easy to prove that the only solution of Catalan equation is (X,Y,p,q) = (2,3,2,3).
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DOI: 10.3923/aja.2009.11.16
URL: https://scialert.net/abstract/?doi=aja.2009.11.16
INTRODUCTION
More than one century after the formulation of the famous conjecture by the Belgian mathematician Eugene Catalan, Catalan Conjecture has been proved by Mihailescu (2004). It is no more an open problem, but it is still interesting for the researchers, because P. Mihailescu has opened the door to other proofs. We propose, in this research, a solution which is a variant of Fermat-Catalan conjecture one. It is based on an Algebraic-Analytic Approach that we have developed for Diophantine Equations.
THE PROOF
Let Catalan equation(1) |
(2) |
(3) |
(4) |
(5) |
(6) |
(7) |
(8) |
The initial hypothesis is false, the only solution is a = b = c = 1
Then
solution is p = 2, which means . Let us prove it
We deduce
Lemma 1
xi, yi have the following expressionsProof of Lemma 1
Lemma 2
For x ≠ yLemma 3
xi-yi = x-y
Lemma 4
The solution of lemmas 2, 3, 4 is
xy(x-y) = 0
Proof of Lemma 4
We conclude that
And there is no solution, it means that p = 2, effectively, the expression of the sequences for p = 1, is
As there is an infinity of solutions for p = 1, the expressions of the sequences imply the existence of solutions for p = 2 and does not guarantee at all the existence of the sequences for p>2 and I = 2. Then, p = 2 and q = 3 is the only solution.
The equation becomes
The only solution of Catalan equation is effectively
(X,Y,p,q) = (2,3,2,3)
CONCLUSION
Catalan equation has effectively only one solution an elementary proof exists. It seems that many open problems of number theory can be solved by the same way. How ? We showed one solution.
REFERENCES
- Mihăilescu, P., 2004. Primary cyclotomic units and a proof of catalan's conjecture. J. Reine Angew. Math., 572: 167-195.
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