Displaying similar documents to “The diophantine equations ( x 2 - c ) 2 = ( t 2 ± 2 ) y 2 + 1 ”

The diophantine equation a x 2 + b x y + c y 2 = N , D = b 2 - 4 a c > 0

Keith Matthews (2002)

Journal de théorie des nombres de Bordeaux

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We make more accessible a neglected simple continued fraction based algorithm due to Lagrange, for deciding the solubility of a x 2 + b x y + c y 2 = N in relatively prime integers x , y , where N ≠ 0 , gcd ( a , b , c ) = gcd ( a , N ) = 1 et D = b 2 - 4 a c > 0 is not a perfect square. In the case of solubility, solutions with least positive y, from each equivalence class, are also constructed. Our paper is a generalisation of an earlier paper by the author on the equation x 2 - D y 2 = N . As in that paper, we use a lemma on unimodular matrices that gives a much simpler proof than Lagrange’s...

The ratio and generating function of cogrowth coefficients of finitely generated groups

Ryszard Szwarc (1998)

Studia Mathematica

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Let G be a group generated by r elements g 1 , … , g r . Among the reduced words in g 1 , … , g r of length n some, say γ n , represent the identity element of the group G. It has been shown in a combinatorial way that the 2nth root of γ 2 n has a limit, called the cogrowth exponent with respect to the generators g 1 , … , g r . We show by analytic methods that the numbers γ n vary regularly, i.e. the ratio γ 2 n + 2 / γ 2 n is also convergent. Moreover, we derive new precise information on the domain of holomorphy of γ(z), the generating function...

On an approximation property of Pisot numbers II

Toufik Zaïmi (2004)

Journal de Théorie des Nombres de Bordeaux

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Let q be a complex number, m be a positive rational integer and l m ( q ) = inf { P ( q ) , P ∈ ℤ m [ X ] , P ( q ) ≠ 0 } , where ℤ m [ X ] denotes the set of polynomials with rational integer coefficients of absolute value ≤ m . We determine in this note the maximum of the quantities l m ( q ) when q runs through the interval ] m , m + 1 [ . We also show that if q is a non-real number of modulus > 1 , then q is a complex Pisot number if and only if l m ( q ) > 0 for all m .

Classification of initial data for the Riccati equation

N. Chernyavskaya, L. Shuster (2002)

Bollettino dell'Unione Matematica Italiana

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We consider a Cauchy problem y ′ ⁢ x + y 2 ⁢ x = q ⁢ x , y ⁢ x x = x 0 = y 0 where x 0 , y 0 ∈ R and q ⁢ x ∈ L 1 loc ⁢ R is a non-negative function satisfying the condition: ∫ - ∞ x q ⁢ t ⁢ d ⁢ t > 0 , ∫ x ∞ q ⁢ t ⁢ d ⁢ t > 0 ⁢  for  ⁢ x ∈ R . We obtain the conditions under which y ⁢ x can be continued to all of R . This depends on x 0 , y 0 and the properties of q ⁢ x .