Displaying similar documents to “More on the distribution of the Fibonacci numbers modulo 5 c .”

On wild ramification in quaternion extensions

G. Griffith Elder, Jeffrey J. Hooper (2007)

Journal de Théorie des Nombres de Bordeaux

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This paper provides a complete catalog of the break numbers that occur in the ramification filtration of fully and thus wildly ramified quaternion extensions of dyadic number fields which contain - 1 (along with some partial results for the more general case). This catalog depends upon the , which as defined in [] is associated with the biquadratic subfield. Moreover we find that quaternion counter-examples to the conclusion of the Hasse-Arf Theorem are extremely rare and can occur only...

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...

Rational approximations to 2 3 and other algebraic numbers revisited

Paul M. Voutier (2007)

Journal de Théorie des Nombres de Bordeaux

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In this paper, we establish improved effective irrationality measures for certain numbers of the form n 3 , using approximations obtained from hypergeometric functions. These results are very close to the best possible using this method. We are able to obtain these results by determining very precise arithmetic information about the denominators of the coefficients of these hypergeometric functions. Improved bounds for the Chebyshev functions in arithmetic progressions θ ( k , l ; x ) and...

Fibonacci numbers and Fermat's last theorem

Zhi-Wei Sun (1992)

Acta Arithmetica

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Let Fₙ be the Fibonacci sequence defined by F₀=0, F₁=1, F n + 1 = F + F n - 1 ( n 1 ) . It is well known that F p - ( 5 / p ) 0 ( m o d p ) for any odd prime p, where (-) denotes the Legendre symbol. In 1960 D. D. Wall [13] asked whether p ² | F p - ( 5 / p ) is always impossible; up to now this is still open. In this paper the sum k r ( m o d 10 ) n k is expressed in terms of Fibonacci numbers. As applications we obtain a new formula for the Fibonacci quotient F p - ( 5 / p ) / p and a criterion for the relation p | F ( p - 1 ) / 4 (if p ≡ 1 (mod 4), where p ≠ 5 is an odd prime. We also prove that the affirmative...