8 Further Remarks 217F(φ(t),ψ(t))is identically zero. The coefficients may be taken fromQif the curve
has at least one non-singular rational point.
Thus in retrospect, and quite unfairly, Siegel’s remarkable result may be seen as
simply picking out those curves of genus 0 which have infinitely many integral points,
a problem which had already been treated by Maillet (1919).
In this connection it may be mentioned that the formula for Pythagorean triples
given in§5 of Chapter II may be derived from the parametrization of the unit circle
x^2 +y^2 =1 by the rational functions
x(t)=( 1 −t^2 )/( 1 +t^2 ), y(t)= 2 t/( 1 +t^2 ).8 FurtherRemarks
More extensive accounts of the theory of continued fractions are given in the books
of Rockett and Szusz [45] and Perron [41]. Many historical references are given in
Brezinski [12]. The first systematic account of the subject, which it is still a delight to
read, was given in 1774 by Lagrange [32] in his additions to the French translation of
Euler’sAlgebra.
The continued fraction algorithm is such a useful tool that there have been many
attempts to generalize it to higher dimensions. Jacobi, in a paper published posthu-
mously (1868), defined a continued fraction algorithm inR^2. Perron (1907) extended
his definition toRnand proved that convergence holds in the following weak sense:
for a given nonzerox∈ Rn, the Jacobi-Perron algorithm constructs recursively a
sequence of basesBk ={bk 1 ,...,bkn}ofZnsuch that, for eachj∈{ 1 ,...,n},the
angle between the lineObkjand the lineOxtends to zero ask→∞. More recently,
other algorithms have been proposed for which convergence holds in the strong sense
that, for eachj ∈{ 1 ,...,n}, the distance ofbkjfrom the lineOxtends to zero as
k→∞. See Brentjes [11], Ferguson [22], Just [28] and Lagarias [31].
Proposition 2 was first proved by Serret [51]. Proposition 3 was proved by
Lagrange. The complete characterization of best approximations is proved in the book
of Perron.
Lambert (1766) proved thatπwas irrational by using a continued fraction expan-
sion for tanx. For the continued fraction expansion ofπ, see Choonget al.[15]. Badly
approximable numbers are thoroughly surveyed by Shallit [52].
The theory of Diophantine approximation is treated more comprehensively in the
books of Koksma [30], Cassels [13] and Schmidt [47].
The estimateO(
√
DlogD)for the period of the continued fraction expansion of a
quadratic irrational with discriminantDis proved by elementary means in the book of
Rockett and Szusz. Further references are given in Podsypanin [42].
The ancient Hindu method of solving Pell’s equation is discussed in Selenius [49].
Tables for solving the Diophantine equationx^2 −dy^2 =m,wherem^2 <d,are
given in Patz [39]. Pell’s equation plays a role in the negative solution of Hilbert’s
tenth problem, which asks for an algorithm to determine whether an arbitrary polyno-
mial Diophantine equation is solvable in integers. See Daviset al.[18] and Jones and
Matijasevic [26].