Does that hyperbol a contai n intege r lattic e points other tha n (±

Perhaps th e curv e carefull y navigate s itsel f s o a s t o mis s al

integer points . A quick search reveals that (8 , 3) satisfies th e eq

and thu s ther e exis t nontrivia l solutions . Ar e ther e others ?

Let's perfor m som e algebrai c gymnastic s wit h ou r new-f

nontrivial solution :

82

- 7(3)

2

= ( 8 - Sy/fj (s + Sy/fj = 1,

and thu s

? - V7 = — — ^ = - = 0.020915

3 3( 8 + 3\/7 )

From th e previou s identit y w e discover tha t | i s impressively c

the irrationa l A/7 , an d therefor e w e see that a n intege r solutio n

equation lead s to a n amazin g rational approximatio n t o a n asso

irrational number . Thi s observatio n illustrate s a surprisin g c

tion betwee n diophantin e approximatio n an d diophantin e equ

This intriguin g interpla y wil l b e a recurrin g them e througho

journey.

The nam e "diophantine " honor s Diophantus , a mathema

who live d i n Alexandri a sometim e betwee n 150 an d 35 0 A.D

phantus is known for his passion for finding integer solutions to

equations, an d h e was the autho r o f the seminal work Arithmetic

fact i t wa s whil e Pierr e d e Ferma t wa s readin g Arithmetica th

was inspired t o jot a note in the small margin which only later b

known a s Fermat' s Las t Theorem . Ver y fe w detail s ar e know n

the persona l lif e o f Diophantu s outsid e o f th e followin g conu

that appeare d i n Greek Anthology fro m 60 0 A.D.:

God granted him to be a boy for the sixth part of

his life, and adding a twelfth part to this, He clothed

his cheeks with down. He lit him the light of wedlock

after a seventh part, and five years after his marriage

He granted him a son. Alas! late-born wretched child;

after attaining the measure of half his father's life,

chill Fate took him. After consoling his grief by this

science of numbers for four years he ended his life.

Perhaps th e curv e carefull y navigate s itsel f s o a s t o mis s al

integer points . A quick search reveals that (8 , 3) satisfies th e eq

and thu s ther e exis t nontrivia l solutions . Ar e ther e others ?

Let's perfor m som e algebrai c gymnastic s wit h ou r new-f

nontrivial solution :

82

- 7(3)

2

= ( 8 - Sy/fj (s + Sy/fj = 1,

and thu s

? - V7 = — — ^ = - = 0.020915

3 3( 8 + 3\/7 )

From th e previou s identit y w e discover tha t | i s impressively c

the irrationa l A/7 , an d therefor e w e see that a n intege r solutio n

equation lead s to a n amazin g rational approximatio n t o a n asso

irrational number . Thi s observatio n illustrate s a surprisin g c

tion betwee n diophantin e approximatio n an d diophantin e equ

This intriguin g interpla y wil l b e a recurrin g them e througho

journey.

The nam e "diophantine " honor s Diophantus , a mathema

who live d i n Alexandri a sometim e betwee n 150 an d 35 0 A.D

phantus is known for his passion for finding integer solutions to

equations, an d h e was the autho r o f the seminal work Arithmetic

fact i t wa s whil e Pierr e d e Ferma t wa s readin g Arithmetica th

was inspired t o jot a note in the small margin which only later b

known a s Fermat' s Las t Theorem . Ver y fe w detail s ar e know n

the persona l lif e o f Diophantu s outsid e o f th e followin g conu

that appeare d i n Greek Anthology fro m 60 0 A.D.:

God granted him to be a boy for the sixth part of

his life, and adding a twelfth part to this, He clothed

his cheeks with down. He lit him the light of wedlock

after a seventh part, and five years after his marriage

He granted him a son. Alas! late-born wretched child;

after attaining the measure of half his father's life,

chill Fate took him. After consoling his grief by this

science of numbers for four years he ended his life.