A piece of the real number lin e
A tiny portio n o f the previou s lin e segment magnifie d
Thus w e se e tha t th e lin e i s suc h a dens e numbe r jungl e tha t
magnification reveal s n o greate r detail . We'l l discove r tha t so
the ric h an d beautifu l structur e o f numbers ca n be brought int
by examinin g ho w wel l irrationa l number s ca n b e approximat
rational ones . Thi s strateg y i s the essenc e o f diophantine appro
tion.
The othe r sid e o f diophantin e analysis , th e stud y o f dioph
equations, revolve s aroun d th e followin g basi c question : Ho w
equation be solved if the variables are to take only integer (o r ra
values? Fo r example , let' s conside r th e equatio n
x2
- 7y
2
= 1.
Clearly (±1,0) ar e solutions, but we'l l consider thes e trivial sol
Are ther e an y nontrivial intege r solutions ? Ar e ther e infinitel y
integer solutions ? I s ther e a n algorith m t o generat e the m all
following grap h depict s th e real solutions t o th e equation .
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