Number Theory: An Introduction to Mathematics

7 Complements 211

It was conjectured by Frobenius (1913) that a Markov triple is uniquely determined
by its greatest element. This has been verified whenever the greatest element does not
exceed 10^140. It has also been proved when the greatest element is a prime (and in
some other cases) by Baragar (1996), using the theory of quadratic fields.

7 Complements...............................................

There is an important analogue of the continued fraction algorithm for infinite series.
LetKbe an arbitrary field and letFdenote the set of all formal Laurent series

f=

∑

n∈Z

αntn

with coefficientsαn∈Ksuch thatαn=0 for at most finitely manyn>0. If

g=

∑

n∈Z

βntn

is also an element ofF, and if we define addition and multiplication by

f+g=

∑

n∈Z

(αn+βn)tn, fg=

∑

n∈Z

γntn,

whereγn=

∑

j+k=nαjβk,thenFacquires the structure of a commutative ring. In
fact,Fis a field. For, iff=

∑

n≤vαnt

n,whereαv=0, we obtaing=∑

n≤−vβnt

n

such thatfg=1 by solving successively the equations

αvβ−v= 1

αvβ−v− 1 +αv− 1 β−v= 0

αvβ−v− 2 +αv− 1 β−v− 1 +αv− 2 β−v= 0

·····

Define the absolute value of an elementf=

∑

n∈Zαnt

nofFby putting

|O|= 0 , |f|= 2 v(f) iff=O,

wherev(f)is the greatest integernsuch thatαn=0. It is easily verified that

|fg|=|f||g|,|f+g|≤max(|f|,|g|),

and|f+g|=max(|f|,|g|)if|f| =|g|.
For any f=

∑

n∈Zαnt

n∈F,let

f=

∑

n≥ 0

αntn, {f}=

∑

n< 0

αntn

denote respectively its polynomial and strictly proper parts. Then|{f}|< 1, and
|f| = |f|if|f|≥1, i.e. iff =O.