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 :
- 7(3)
= ( 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
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.
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