Number Theory: An Introduction to Mathematics

206 IV Continued Fractions and Their Uses Thus properly equivalent forms have the same discriminant. As the name implies, proper ...

5 The Modular Group 207 denote the number of primitive positive definite quadratic forms with discriminantD which are properly i ...

208 IV Continued Fractions and Their Uses 6 Non-EuclideanGeometry..................................... There is an important con ...

6 Non-Euclidean Geometry 209 obtained by composing such a transformation with the (orientation-reversing) trans- formationx+iy→− ...

210 IV Continued Fractions and Their Uses Then ξn=[an,an+ 1 ,...], − 1 /ηn=[an− 1 ,an− 2 ,...], andξn> 1 ,− 1 <ηn<0. Mo ...

7 Complements 211 It was conjectured by Frobenius (1913) that a Markov triple is uniquely determined by its greatest element. Th ...

212 IV Continued Fractions and Their Uses Iff 0 :=fis not the formal Laurent series of a rational function, we can write f 0 =a ...

7 Complements 213 Proposition 14Let f be a formal Laurent series with convergents pn/qnand let p,q be polynomials with q=O. (i) ...

214 IV Continued Fractions and Their Uses A complex numberζis said to be analgebraic number,orsimplyalgebraic, of degree dif it ...

7 Complements 215 where‖x‖=max(|x 1 |,...,|xn|), is contained in some subspaceVi, except for finitely many points whose number m ...

216 IV Continued Fractions and Their Uses wherec 2 =(c 1 /|a 0 |)^1 /n.Ifk=j,then |x−ζky|≥|(ζj−ζk)y|−|x−ζjy| ≥c 3 |y|−c 2 |y|m/ ...

8 Further Remarks 217 F(φ(t),ψ(t))is identically zero. The coefficients may be taken fromQif the curve has at least one non-sing ...

218 IV Continued Fractions and Their Uses The continued fraction construction for the representation of a primep≡1mod4 as a sum ...

8 Further Remarks 219 M= ( AB CD ) issymplectic,i.e.ifMtJM=J,where J= ( OI −IO ) , then the linear fractional transformationZ→(A ...

220 IV Continued Fractions and Their Uses of the origin and f(z)=λz+O(z^2 ),whereλ =e^2 πiθfor some irrationalθ.It is readily sh ...

9 Selected References 221 [18] M. Davis, Y. Matijasevic andJ. Robinson, Hilbert’s tenth problem. Diophantine equations: positive ...

222 IV Continued Fractions and Their Uses [47] W.M. Schmidt,Diophantine approximation, Lecture Notes in Mathematics 785 , Spring ...

V Hadamard’s Determinant Problem............................. It was shown by Hadamard (1893) that, if all elements of ann×nmatr ...

224 V Hadamard’s Determinant Problem has, ifδ 2 =α 11 α 22 −α 12 α 21 is nonzero, the unique solution ξ 1 =(β 1 α 22 −β 2 α 12 ) ...

1 What is a Determinant? 225 theory. The diffusion of this theory throughout the mathematical world owes much to the clear expos ...