Number Theory: An Introduction to Mathematics

7 Complements 213

Proposition 14Let f be a formal Laurent series with convergents pn/qnand let p,q
be polynomials with q=O.

(i)If|q|<|qn+ 1 |and p/q=pn/qn,then

|qf−p|≥|qn− 1 f−pn− 1 |=|qn|−^1.

(ii)If|qf−p|<|q|−^1 ,then p/q is a convergent of f.

Proof (i) Assume on the contrary that|qf−p|<|qn|−^1 .Since

qn(qf−p)−q(qnf−pn)=qpn−pqn=O

and|qn||qf−p|<1, we must have

|q||qn+ 1 |−^1 =|q||qnf−pn|=|qpn−pqn|≥ 1 ,

which is contrary to hypothesis.
(ii) Assume thatp/qis not a convergent of f.Iff =pN/qNis a rational function
then|q|<|qN|,since

1 ≤|qpN−pqN|=|qf−p||qN|<|q|−^1 |qN|.

Thus, whether or notfis rational, we can choosenso that|qn|≤|q|<|qn+ 1 |. Hence,
by (i),

|qf−p|≥|qn|−^1 ≥|q|−^1 ,

which is a contradiction.

It was shown by Abel (1826) that, for any complex polynomialD(t)which is not a
square, the ‘Pell’ equationX^2 −D(t)Y^2 =1 has a solution in polynomialsX(t),Y(t)
of positive degree if and only if

√

D(t)may be represented as a periodic continued
fraction:

√

D(t)=[a 0 ,a 1 ,...,ah], whereah= 2 a 0 andai=ah−i(i= 1 ,...,h− 1 )
are polynomials of positive degree. By differentiation one obtains

XX′/Y=Y′D+( 1 / 2 )YD′.

It follows thatYdividesX′,sinceXandYare relatively prime, and

(X+Y

√

D)′=(X+Y

√

D)X′/Y

√

D.

Thus the ‘abelian’ integral
∫
X′(t)dt/Y(t)

√

D(t)

is actually the elementary function log{X(t)+Y(t)

√

D(t)}.
Some remarkable results have recently been obtained on the approximation of alge-
braic numbers by rational numbers, which deserve to be mentioned here, even though
the proofs are beyond our scope.