have trie d t o provid e a t th e en d o f eac h chapte r enoug h example s
and experiment s fo r student s t o try , bu t I' m sur e tha t enterprisin g
students an d teacher s wil l b e abl e t o inven t man y more .
What bega n a s a n experimen t i n th e NY U cours e turne d int o
a substantia l revisio n o f th e course . Th e experimen t wa s t o see ho w
much o f number theor y coul d b e formulated i n terms o f "numbers " i n
the mos t primitiv e sense—th e number s 0 , 1, 2 , .. . use d i n counting .
To m y surprise , I foun d tha t no t onl y coul d I avoid negativ e num -
bers bu t tha t I didn't miss them . Th e simpl e reaso n fo r thi s i s tha t
the basi c question s o f numbe r theor y ca n b e state d i n term s o f con -
gruences, an d subtractio n i s alway s possibl e i n congruence s withou t
any nee d fo r negativ e numbers . Negativ e number s hav e alway s le d t o
metaphysical conundrums—wh y shoul d a negativ e time s a negativ e
be a positive?—which caus e confusin g distraction s righ t a t th e outse t
when th e meanin g o f "number " i s bein g mad e precise . I n thi s book ,
the meaning o f "number " derive s simply from th e activit y o f countin g
and arithmeti c ca n begi n immediately . Kronecker' s famou s dictum ,
"God create d th e whole numbers; al l the res t i s human work, " ca n b e
amended t o say , "nonnegativ e whol e numbers, " whic h i s ver y likel y
what Kronecke r mean t anyway .
A central them e o f the boo k i s the proble m I denote b y th e equa -
tion AD + B = , th e proble m o f finding , fo r tw o give n number s A
and B, al l number s x fo r whic h Ax
2
+ B i s a square . A s Chapte r
2 explains , version s o f thi s proble m ar e a t leas t a s ol d a s Pythago -
ras, althoug h tw o millennia late r th e Disquisitiones Arithmeticae stil l
dealt wit h it . A simpl e algorith m fo r th e complet e solutio n i s give n
in Chapte r 19.
Work o n problem s o f th e for m A\D + B = le d Leonhar d Eule r
to th e discover y o f wha t I cal l "Euler' s law, " th e statemen t tha t th e
answer t o th e questio n "I s A a squar e mo d p? " fo r a prim e numbe r
p depends onl y o n th e valu e o f p mod 4A. Thi s statement , o f whic h
the la w o f quadrati c reciprocit y i s a byproduct, i s completel y prove d
in Chapte r 29 .
When Erns t Eduar d Kumme r first introduce d hi s theory o f "idea l
complex numbers " i n 1846, 4 5 year s afte r th e publicatio n o f Disqui-
sitiones Arithmeticae, Gaus s sai d tha t h e ha d worke d ou t somethin g
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