On explicit formulas for the number of solutions to the equation in a finite field.
Baulina, Yu.N. (2003)
Sibirskij Matematicheskij Zhurnal
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Baulina, Yu.N. (2003)
Sibirskij Matematicheskij Zhurnal
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Darvasi, Gyula (2001)
Acta Mathematica Academiae Paedagogicae Nyí regyháziensis. New Series [electronic only]
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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 (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...
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...
Lazareva, O.A. (2008)
Sibirskij Matematicheskij Zhurnal
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Linke, Yu.Yu., Sakhanenko, A.I. (2001)
Sibirskij Matematicheskij Zhurnal
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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 , 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 and...
Zhi-Wei Sun (1992)
Acta Arithmetica
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Let Fₙ be the Fibonacci sequence defined by F₀=0, F₁=1, . It is well known that for any odd prime p, where (-) denotes the Legendre symbol. In 1960 D. D. Wall [13] asked whether is always impossible; up to now this is still open. In this paper the sum is expressed in terms of Fibonacci numbers. As applications we obtain a new formula for the Fibonacci quotient and a criterion for the relation (if p ≡ 1 (mod 4), where p ≠ 5 is an odd prime. We also prove that the affirmative...