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absolutely convergent algebraic integer algebraic number Appendix arbitrarily small best approximations Chapter clearly conjugate algebraic conjugates constant continued fraction convergent convex coprime Corollary defined denote depends Diophantine equations equations equivalent exists finite number fk(z Fm(x follows at once GAUSS'S LEMMA Hence hypothesis implies inequalities infinitely many integer infinitely many solutions integer co-ordinates integer solutions integers q integers x integers xv integral vectors irrational Kronecker's theorem Lemma 11 Lemma 9 lim inf linear forms linearly independent Lj(x Markoff form Minkowski necessary and sufficient notation number of integer number of solutions pairs of integers polynomial positive integers Proof of Theorem proves the lemma PV-number qd—p qn+1 quadratic quadratic form rational numbers real numbers recurrence relation region respectively right-hand side satisfy set of numbers Similarly soluble sufficient condition Theorem III trivial uniformly distributed modulo unit cube values variables volume write Wronskian