Displaying similar documents to “On a theorem of Mestre and Schoof”

A note on linear perturbations of oscillatory second order differential equations

Renato Manfrin (2010)

Archivum Mathematicum

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Under suitable hypotheses on γ ( t ) , λ ( t ) , q ( t ) we prove some stability results which relate the asymptotic behavior of the solutions of u ' ' + γ ( t ) u ' + ( q ( t ) + λ ( t ) ) u = 0 to the asymptotic behavior of the solutions of u ' ' + q ( t ) u = 0 .

Almost powers in the Lucas sequence

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 ( F n ) n ≥ 0 and in the Lucas sequence ( L n ) n ≥ 0 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 L n = q a y p , with a > 0 and p ≥ 2 , for all primes q < 1087 and indeed for all but 13 primes q < 10 6 . Here the strategy of [] is not sufficient due to the sizes...

Weber’s class number problem in the cyclotomic ℤ 2 -extension of ℚ , II

Takashi Fukuda, Keiichi Komatsu (2010)

Journal de Théorie des Nombres de Bordeaux

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Let h n denote the class number of n -th layer of the cyclotomic ℤ 2 -extension of ℚ . Weber proved that h n ( n ≥ 1 ) is odd and Horie proved that h n ( n ≥ 1 ) is not divisible by a prime number ℓ satisfying ℓ ≡ 3 , 5 ( mod 8 ) . In a previous paper, the authors showed that h n ( n ≥ 1 ) is not divisible by a prime number ℓ less than 10 7 . In this paper, by investigating properties of a special unit more precisely, we show that h n ( n ≥ 1 ) is not divisible by a prime number ℓ less than 1 . 2 · 10 8 . Our argument also leads to the conclusion that h n ( n ≥ 1 ) is not divisible by...