imensions is given by G. Cantor in vol. 2 of the
_Acta Mathematica_.

Applications of simple continued fractions to the theory of numbers, as,
for example, to prove the theorem that a divisor of the sum of two
squares is itself the sum of two squares, may be found in J. A. Serret's
_Cours d'Algebre Superieure_.

2. _Recurring Simple Continued Fractions._--The infinite continued
fraction

1 1 1 1 1 1 1 1 1 1
a1 + -- -- --- -- -- --- -- -- --- --
a2 + a3 ... + a_n + b1 + b2 ... + b_n + b1 + b2 ... + b_n + b1 + ...,

where, after the n^{th} partial quotient, the cycle of partial quotients
b1, b2, ..., b_n recur in the same order, is the type of a recurring
simple continued fraction.

The value of such a fraction is the positive root of a quadratic
equation whose coefficients are real and of which one root is negative.
Since the fraction is infinite it cannot be commensurable and therefore
its value is a quadratic surd number. Conversely every positive
quadratic surd number, when expressed as a simple continued fraction,
will give rise to a recurring fraction. Thus

__ 1 1 1 1 1
2 - \/ 3 = -- -- -- -- --
3 + 1 + 2 + 1 + 2 + ...,

___ 1 1 1 1 1 1 1 1
\/ 28 = 5 + -- -- -- -- -- -- -- --
3 + 2 + 3 + 10 + 3 + 2 + 3 + 10 + ...

The second case illustrates a feature of the recurring continued
fraction which represents a complete quadratic surd. There is only one
non-recurring partial quotient a1. If b1, b2, ..., b_n is the cycle of
recurring quotients, then b_n = 2a1, b1 = b_{n-1}, b2 = b_{n-2}, b3 =
b_{n-3}, &c.

In the case of a recurring continued fraction which represents [sqr]N,
where N is an integer, if n is the number of partial quotients in the
recurring cycle, and p_{nr}/q_{nr} the nr^{th} convergent, then p^2_{nr}
-Nq^2_{nr} = (-1)^{nr}, whence, if n is odd, integral solutions of the
indeterminate equation x squared - Ny squared = +-1 (the so-called Pellian equation)
can be found. If n is even, solutions of the equation x squared -Ny squared = +1 can
be found.

The theory and development of the simple recurring continued fraction is
due to Lagrange. For proofs of the theorems here stated and for
applications to the more general indeterminate equation x squared -Ny squared = H the
reader may consu