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Tečny dvou kruhů

Rajmund Fišer (1907)

Časopis pro pěstování mathematiky a fysiky

The Büchi sequences and Hilbert's Tenth Problem

(2016)

Banach Center Publications

In this short survey paper we state the Büchi conjecture and discuss its relations with the Hilbert Tenth Problem. We give some generalizations of the conjecture, and include some numerical examples.

The classification of pairs of binary quadratic forms

Jorge Morales (1991)

Acta Arithmetica

The D(1)-extensions of D(-1)-triples 1,2,c and integer points on the attached elliptic curves

Yasutsugu Fujita (2007)

Acta Arithmetica

The Diophantine equation $a x^{2} + 2 b x y - 4 a y^{2} = \pm 1$ .

Bapoungué, Lionel (2003)

International Journal of Mathematics and Mathematical Sciences

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

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 $N \neq 0$ , 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 for...