ALGEBRA I N ANCIEN T AN D MODER N TIME S 5
The sequenc e o f primes i s infinite .
The proo f i s very simpl e an d ye t remarkabl y beautiful . I n th e exercise s belo w w e
shall sketch th e argumen t give n i n Euclid .
Because o f this resul t i t i s clear tha t ther e mus t b e enormously larg e number s
which are primes. Bu t i f a very large number i s given, it is not a t al l an easy matte r
to decid e i f i t i s a prime . Unbelievably , thi s proble m require s ver y sophisticate d
methods for it s solution. Eve n in early times large primes were a source of interest .
For instance , Ferma t believe d tha t al l numbers o f the for m
An = 2 2n + 1, n = 0,l,2,3,.. .
are primes . Thi s i s tru e fo r n upt o 4 bu t fals e fo r n = 5 a s Eule r verifie d later .
Nowadays testing for primality as well as factorization o f large numbers is attracting
a lo t o f attentio n becaus e o f th e relationshi p o f thes e question s t o th e proble m o f
constructing an d breakin g codes .
NOTES AN D EXERCISE S
I hav e mentione d tha t Euclid' s axiom s ar e rathe r subtl e i n man y ways . Fo r instance , th e
plane i n Euclid' s geometr y i s no t th e surfac e o f th e eart h bu t a n idea l plan e wher e line s stretc h
out t o infinit y i n bot h directions . Thi s i s rathe r remarkable , becaus e eve n thoug h th e origin s o f
geometry g o back ultimatel y t o measurement s carrie d ou t o n the surfac e o f the earth, nevertheless ,
when Eucli d idealize d them , everythin g too k plac e i n th e idea l plane .
Among Euclid' s axiom s i s th e so-calle d parallel postulate, whic h stoo d ou t fro m th e res t
because i t wa s ver y har d t o accep t i t o n an y intuitiv e basis . Thi s postulat e amount s t o sayin g
that i f a lin e I i s give n an d a poin t P no t o n i t i s als o given , ther e i s exactly on e lin e throug h P
which does not mee t L Eve n Eucli d mus t hav e been uncomfortabl e wit h i t a s may b e seen from th e
fact tha t h e mad e a n effor t t o prov e man y proposition s withou t th e ai d o f th e paralle l postulate ,
and starte d usin g i t onl y whe n i t becam e absolutel y necessary . Hi s successor s trie d i n vai n t o
deduce thi s axio m fro m th e othe r axioms , an d thi s effort , lastin g severa l centuries , ultimatel y le d
to th e creation o f non euclidean geometry, b y Bolyai (1802-1860) an d Lobachevsk y (1793-1856).
After th e publicatio n o f Euclid' s Elements, th e axiomati c approac h becam e th e preferre d
way t o develo p an y mathematica l topic . I t wa s fel t tha t th e axiomati c metho d represente d a n
ideal of perfection, an d tha t th e succes s o f any developmen t wa s measure d agains t tha t o f Euclid' s
geometry. A s mentioned before , eve n Euclid' s axiom s ha d t o b e enlarge d an d modifie d s o that th e
deductions mad e b y hi m wer e permissible . Th e mai n figure s i n thi s lin e o f though t wer e Pasc h
(1843-1930) an d Hilber t (1862-1943). Nowaday s th e axiomati c metho d ha s penetrate d al l part s
of mathematic s an d i s th e onl y approac h used .
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