On the positive integral solutions of the Diophantine equation .
Luca, Florian, Togbé, Alain (2008)
Bulletin of the Malaysian Mathematical Sciences Society. Second Series
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Luca, Florian, Togbé, Alain (2008)
Bulletin of the Malaysian Mathematical Sciences Society. Second Series
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Filipin, Alan, Togbé, Alain (2009)
Acta Mathematica Academiae Paedagogicae Nyí regyháziensis. New Series [electronic only]
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Harald A. Helfgott (2007)
Journal de Théorie des Nombres de Bordeaux
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Let be a polynomial of degree without roots of multiplicity or . Erdős conjectured that, if satisfies the necessary local conditions, then is free of th powers for infinitely many primes . This is proved here for all with sufficiently high entropy. The proof serves to demonstrate two innovations: a strong repulsion principle for integer points on curves of positive genus, and a number-theoretical analogue of Sanov’s theorem from the theory of large deviations. ...
Yoshinobu Nakai, Iekata Shiokawa (1992)
Acta Arithmetica
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Nedeljkov, M., Pilipović, S. (2002)
Bulletin. Classe des Sciences Mathématiques et Naturelles. Sciences Mathématiques
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M. Nedeljkov, S. Pilipović (2002)
Bulletin, Classe des Sciences Mathématiques et Naturelles, Sciences mathématiques
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Yann Bugeaud, Florian Luca, Maurice Mignotte, Samir Siksek (2008)
Journal de Théorie des Nombres de Bordeaux
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The famous problem of determining all perfect powers in the Fibonacci sequence and in the Lucas sequence has recently been resolved []. The proofs of those results combine modular techniques from Wiles’ proof of Fermat’s Last Theorem with classical techniques from Baker’s theory and Diophantine approximation. In this paper, we solve the Diophantine equations , with and , for all primes and indeed for all but primes . Here the strategy of [] is not sufficient due to the sizes...
Begehr, Heinrich (2005)
Boletín de la Asociación Matemática Venezolana
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Jerzy Browkin (1999)
Colloquium Mathematicae
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T. Cochrane and R. E. Dressler [CD] proved that the abc-conjecture implies that, for every > 0, the gap between two consecutive numbers A with two exceptions given in Table 2.