ing abou t th e compositio n o f binar y quadrati c form s i n Sectio n 5 of
Disquisitiones Arithmeticae, bu t tha t h e lef t i t ou t o f th e boo k be -
cause h e ha d no t bee n abl e t o pu t i t o n firm ground.
2
Althoug h th e
proof o f quadrati c reciprocit y give n i n thi s boo k wa s originall y in -
spired b y Gauss' s proo f usin g th e compositio n o f forms , i t i s state d
in term s close r t o Rummer' s idea l numbers . Specifically :
If, i n additio n t o usin g ordinar y number s 0 , 1, 2 , .. . , on e com -
putes wit h a symbo l \J~A whos e squar e i s a fixed numbe r A , on e ha s
an arithmetic— I hav e dubbe d i t th e arithmeti c o f "hypernumbers "
for tha t A —in whic h th e natura l generalizatio n o f doin g computa -
tions mo d n fo r som e numbe r n i s t o d o computation s mo d [a , b] for
some pair o f hypernumber s a an d b. (Wit h ordinar y numbers , th e
Euclidean algorith m serve s t o reduc e th e numbe r o f number s i n a se t
that describe s a modulus to just one , but wit h hypernumbers tw o may
be needed , a s i s show n i n Chapte r 18.) Wit h natura l definition s o f
multiplication an d equivalenc e o f suc h "module s o f hypernumbers, "
the computations neede d t o solve AD + B = an d t o prove quadrati c
reciprocity ca n b e explaine d ver y simply . I n thi s way , Gauss' s diffi -
cult compositio n o f form s i s avoide d bu t th e essenc e o f hi s metho d i s
preserved.
The las t tw o chapter s relat e th e method s o f th e boo k t o Gauss' s
binary quadrati c form s s o student s intereste d i n readin g furthe r i n
the Disquisitiones Arithmeticae —or student s interested i n binary qua -
dratic forms—wil l b e abl e t o mak e th e transition .
Finally, a n appendi x give s a table o f the cycle s o f stabl e module s
of hypernumbers fo r al l numbers A 111 that ar e not squares , whic h
will b e usefu l fo r students , a s they wer e fo r me , i n understandin g th e
general theor y an d i n workin g ou t examples .
2
See
[E4].
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