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Элементарная теория сумм Гаусса

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Yeaiaioa?iay oai?ey noii Aaonna.

?anniio?ei neaaeothuoth noiio – noiio Aaonna :

aaea D – oeaeia iieiaeeoaeueiia e (a, D)=1.

Iieaaeai, /oi cia/aiea noiiu aoaeao iaeiei e oai aea, anee o
i?iaaaaao ethaoth iieioth nenoaio au/aoia ii iiaeoeth D.

Aeaenoaeoaeueii, ionoue o i?iaaaaao iieioth nenoaio au/aoia ii iiaeoeth
D. Oiaaea o=qD+k , aaea k =0, 1, …, D-1 , q ?
Z

Aoaeai eiaoue :

/oi e o?aaiaaeinue.

Eaiia 1.

Ionoue (a, D)=1. Oiaaea:

Aeieacaoaeuenoai:

Ii naienoao iiaeoey eiiieaeniiai /enea :

-2-

Eiaai:

Naeaeaai caiaio x = x + t . Eiaaea o e o i?iaaaatho iieioth
nenoaio au/aoia ii iiaeoeth D , io o e t i?iaaaatho iacaaeneii
iieiua nenoaiu au/aoia ii iiaeoeth D.

Aeaenoaeoaeueii, ionoue o e o i?iaaaatho iieioth nenoaio au/aoia ii
iiaeoeth D . Oiaaea o = qD + k k=0,
1, …, D-1 , q ? Z

o = pD + i
i=0, 1, …, D-1 , p ? Z

Neaaeiaaoaeueii, t = x – x = (q – p)D + (k – i) = l D + m , aaea m=0,
1, …, D-1 , l ? Z

a) Ionoue D – ia/aoiia, o.a. (2a, D)=1

anee D aeaeeo t.

Anee aea D ia aeaeeo t, oi iineaaeithth noiio iiaeii caienaoue a aeaea
:

Iieo/eee :

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Oiaaea

Ionthaea

a) Ionoue D aeaeeony ia 4, o.a. aiciiaeii i?aaenoaaeaiea : D = 2D ,
aaea D – /aoiia e ( a, D )=1 .

Iieo/ei :

Oae eae D /aoiia, oi

Neaaeiaaoaeueii

a) Ionoue D = 2 (mod 4) , o.a. D = 4q + 2 , q ? Z

Oiaaea ec i?aaeuaeouaai neo/ay eiaai : D = 2 (2q+1)= 2D , D –
ia/aoiia. Eiaai :

*oi e o?aaiaaeinue.

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Eaiia 2.

Anee D e D acaeiii i?inoua /enea, oi

S ( aD1 , D2 ) S ( aD2 , D1 ) = S ( a , D1 D2 )

Aeieacaoaeuenoai:

A yoeo noiiao t1 i?iaaaaao iieioth nenoaio au/aoia ii iiaeoeth D2 , a
t2 i?iaaaaao iieioth nenoaio au/aoia ii iiaeoeth D2. I?e yoii D1t1
+ D2t2 i?iaaaaao iieioth nenoaio au/aoia ii iiaeoeth D1D2 .
Aeaenoaeoaeueii , anaai /eaiia a noiia D1D2 e ieeaeea aeaa
ian?aaieiu iaaeaeo niaie. Aeaenoaeoaeueii, i?aaeiieiaeei i?ioeaiia :
ionoue D1t1 + D2t2 = D1t1 + D2t2 ( mod D1D2 )

Ionthaea D1 (t1 – t1) = D2 (t2 – t2 ) (mod D1D2) Oiaaea

D1 (t1 – t1) = D2 (t2 – t2 ) (mod D2) A oae eae
D2 (t2 – t2 ) = 0 (mod D2)

Oi ii naienoao n?aaiaiee eiaai D1 (t1 – t1) = 0 (mod D2)
Ionthaea oae eae (D1, D2)=1 , oi t1 – t1 = 0 (mod D2)
Aiaeiae/ii iieo/ei t2 – t2 = 0 (mod D1)

O.a. eiaai t1 = t1 (mod D2) e t2 = t2 (mod D1)
. Ii yoi i?ioeai?a/eo oiio, /oi t1 i?iaaaaao iieioth nenoaio au/aoia
ii iiaeoeth D2 , a t2 i?iaaaaao iieioth nenoaio au/aoia ii iiaeoeth
D2, oae eae a iieiie nenoaia au/aoia ethaua aeaa /enea ia n?aaieiu.
Neaaeiaaoaeueii iaoa i?aaeiieiaeaiea auei iaaa?iui e aeaenoaeoaeueii
D1t1 + D2t2 i?iaaaaao iieioth nenoaio au/aoia ii iiaeoeth D1D2 .

Iiyoiio

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Eaiia 3.

Ionoue p i?inoia ia/aoiia /enei e ia aeaeeo a . Oiaaea

Aeieacaoaeuenoai:

/oi e o?aaiaaeinue aeieacaoue.

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Eaiia 4.

Anee ? i?inoia ia/aoiia /enei , oi

Aeieacaoaeuenoai :

Ec eaiiu 3. iieo/ei

Oae eae i?iecaaaeaiea nii?yaeaiiuo aaee/ei aeaao eaaae?ao iiaeoey, oi

Eaiia 5.

Anee ? e q ?acee/iua i?inoua /enea , oi

Aeieacaoaeuenoai :

Oae eae ( ?, q )= 1 , iu iiaeai ainiieueciaaoueny eaiiie 2 : a iaoai
neo/aa

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Eoae , iu iieacaee, /oi

/oi e o?aaiaaeinue aeieacaoue.

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