CC BY
3TAQQOSLAMALAR .EYLER FUNKSIYASI.
Sharipova Madina Po'latovna
Osiyo Xalqaro Universiteti "Umumtexnik fanlar" kafedrasi o'qituvchisi [email protected] Latipova Shahnoza Salim qizi
Osiyo Xalqaro Universiteti "Umumtexnik fanlar" kafedrasi o'qituvchisi [email protected]
ARTICLE INFO
Qabul qilindi: 10-February 2024 yil Ma'qullandi: 15- February 2024 yil Nashr qilindi: 22- February 2024 yil
KEY WORDS
Butun sonlar halqasi,chegirmalar sinfi,modul,Eyler funksiyasi,Ferma teoremasi..
ABSTRACT
Maqolada Taqqoslamalar ularning xosslari o'rganilgan.TaqqosIamalarni Eyler va Ferma teoremalari orqali o'rganish.
Z-butun sonlar halqasi bo'lib, m>1 natural son bo'lsin. Ta'rif. Agar Z halqaga tegishli a va b
sonlarni m natural songa bo'lganda hosil bolgan qoldiqlar bir xil bo'lsa, yoki a-b ayirma m ga
bo'linsa, yoki a=b+mq tenglik o'rinli bo'lsa, u holda a va b sonlar m modul bo'yicha
taqqoslanadi deyiladi va uni a=b(mod m) ko'rinishda belgilanadi.
Taqqoslamalar quyidagi xossalarga ega:
10. Taqqoslama ekvivalent binar munosabat.
20.Bir xil modulli taqqoslamalarni hadma-had qo'shish (ayirish) mumkin. Bu ish n ta ai=bi(mod m), a2=b2(mod m),...,an=bn (mod m) taqqoslamalar uchun ham bajariladi, ya'ni ai±a2±...± an=(bi±b2±...±bn) (mod m) taqqoslamani hosil qilamiz.
Natija. Taqqoslamaning bir qismidagi sonni uning ikkinchi qismiga qarama-qarshi ishora bilan o'tkazish mumkin.
Natija. Taqqoslamaning ixtiyoriy qismiga modulga karrali sonni qo'shish mumkin. 30. Bir xil modulli taqqoslamalarni hadma-had ko'paytirish mumkin.
Natija. Taqqoslamaning ikki qismini (modulni o'zgartirmay) bir xil natural darajaga ko'tarish mumkin.
40. Modulni o'zgartirmagan holda taqqoslamaning ikki qismini bir xil butun songa ko'paytirish mumkin.
50.Agar x=y(mod m) bolsa, u holda ixtiyoriy butun koeffitsientli f(x)=aoxn+aixn-1+... +an-ix+an, f(y)=aoyn+aiyn-1+...+an-iy+an ko'phadlar uchun f(x)=f(y) (mod m) taqqoslama o>inli bo'ladi.
60.Agar bir vaqtda apbi (mod m)(i= 1, n ) va x= y (mod m) taqqoslamalar o>inli bolsa, u holda
ao xn+ai xn-1l+...+an-ix +an = bo yn + bi yn-1 +...+bn-i y+bn(mod m) taqqoslama o'rinli bo'ladi.
Natija. Taqqoslamada qatnashuvchi qo'shiluvchini o'zi bilan teng qoldiqli bo'lgan ikkinchi songa almashtirish mumkin.
70. Taqqoslamaning ikki qismini modul bilan o'zaro tub bo'lgan ko'paytuvchiga qisqartirish mumkin.
80. Taqqoslamaning ikki qismini va modulini bir xil musbat songa ko'paytirish, taqqoslamaning ikki qismi va moduli umumiy ko'paytuvchiga ega bo'lsa, u xolda bu taqqoslamaning ikki qismi va modulini bu umumiy ko'paytuvchiga boyish mumkin.
90. Agar taqqoslama bir necha Modul bo'yicha o'rinli bo'lsa, u holda bu taqqoslama shu modullarning eng kichik umumiy bo'linuvchisi bo'yicha ham o'rinli bo'ladi.
100. Agar taqqoslama biror m Modul bo'yicha o'rinli bo'lsa, u holda bu takdoslama modulning ixtiyoriy buluvchisi buyicha ham o'rinli bo'ladi.
ii0. Taqqoslamaning bir qismi va modulining EKUB bilan uning ikkinchi qismi va modulining EKUB o'zaro teng bo'ladi.
Barcha butun sonlarni m>l natural songa bo'lganda 0, 1, 2, ..., m-1 qoldiqlar hosil bo'ladi. Bunday har bir qoldiqqa butun sonlarning biror sinfi mos keladi. Ta'rif. m ga bo'linganda r ga teng bir xil qoldiq beradigan butun sonlar to'plami m modul
bo'yicha chegirmalar sinflari deyiladi va uni r kabi belgilanadi.
Ta'rif. Chegirmalar sinfining ixtiyoriy elementi shu sinfning chegirmasi deyiladi.
Ta'rif. m Modul bo'yicha tuzilgan har bir chegirmalar sinfidan erkinlik bilan bittadan element
olib tuzilgan to'plamga m Modul bo'yicha chegirmalarning to'la sistemasi deyiladi.
Sinfning bitta chegirmasi m Modul bilan o'zaro tub bo'lsa, u holda bu sinfning barcha
elementlari ham m Modul bilan o'zaro tub bo'ladi.
• ^ I W l^H
Ta'rif. m Modul bilan o'zaro tub bo'lgan barcha chegirmalar sinfidan erkinlik bilan bittadan
chegirma olib tuzilgan to'plam chegirmalarning m Modul bo'yicha keltirilgan sistemasi deyiladi.
m modul bo'yicha chegirmalarning keltirilgan sistemasidagi elementlar sonini aniqlash uchun Eyler funktsiyasi deb ataluvchi ^(m) funksiyadan foydalanamiz.
Ta'rif. Agar quyidagi ikkita shart bajarilsa, u holda ^(m) sonli funktsiya Eyler funktsiyasi deyiladi: 1. 9(1) = 1.
2. ^(m) funktsiya m dan kichik va m bilan o'zaro tub bo'lgan natural sonlar soni. Ta'rif. Natural sonlar to'plamida aniqlangan f funktsiya uchun (m; n)=1 bo'lganda f(m-n)=f(m)-f(n) tenglik bajarilsa, u holda f funktsiyaga mul'tiplikativ funktsiya deyiladi.
Teorema. Eyler funktsiyasi mul'tiplikativ funktsiya bo'ladi.^(m) Eyler funksiyasini hisoblash formulalari quyidagilardan iborat: m=p tub son bo'lsa, u holda ^(p)=r-1 bo'ladi. m= ra (r-tub son, a-natural son) bo'lsa, u holda 9(pa)=pa-1-(p-1) bo'ladi.
m= Pia1 P 2 a2-PK '
bo'lsa, u holda
9(m) = 9(PiaiP2 a2-Pk ak )= m
1 -
P
i y
1-
P
2 y
1-
Pk
K
1
1
1
o'ladi.
Eyler teoremasi. Agar (a; m)=1 bo'lsa, u holda a9(m)=1(mod m) taqqoslama o'rinli bo'ladi.
Ferma teoremasi. Agar a son r tub songa bo'linmasa, u holda ap-1=1 (mod m) taqqoslama o'rinli bo'ladi.
Koeffitsientlari butun sonlardan iborat f(x)= ao xn+ +ai- •x"-1 ...an-ix+an ko'phad berilgan bo'lsin.
Ta'rif. Ushbu
f(x)=0(mod m) (ao son m ga bo'linmaydi, a*eZ, m>1) (1)
ko'rinishdagi taqqoslamani bir noma'lumli n- darajali taqqoslama deyiladi.
Ta'rif. Agar x=s bo'lganda
f(c)=0(mod m) (2)
taqqoslama to'g'ri bo'lsa, u holda s son (1) taqqoslamani qanoatlantiradi deyiladi.
Teorema. Agar s son (1) taqqoslamani qanoatlantirsa, u holda C chegirmalar sinfiga tegishli ixtiyoriy son ham (1) taqqoslamani qanoatlantiradi.
Ta'rif. Agar s son (1) taqqoslamani qanoatlantirsa, u holda C chegirmalar sinfi (1) taqqoslamaning echimi deyiladi.
m modul bo'yicha barcha chegirmalar sinfi 0,1,2,...,m-1 bo'ladi. Demak, m modulli taqqoslamani qanoatlantiruvchi sonlarni 0,1,2,..., m-1 sonlar ichidan qidirish lozim.
Ta'rif. Yechimlari to'plami ustma-ust tushgan taqqoslamalarni teng kuchli taqqoslamalar deyiladi.
Agar (1) taqqoslamaning ikki qismiga ixtiyoriy ko'phad qo'shilsa yoki har ikki qismini m Modul bilan o'zaro tub bo'lgan k songa ko'paytirilsa, yoki ikki qismi va modulini k natural songa ko'paytirilsa, u holda hosil bo'lgan taqqoslama berilgan taqqoslamaga teng kuchli bo'ladi.
Ta'rif. Ushbu ax=b(mod m) (a,beZ,VmeN) (3)
ko'rinishdagi taqqoslamaga bir noma'lumli birinchi darajali taqqoslama deyiladi.
Teorema. Agar (a;m)=1 bo'lsa, u holda (3) taqqoslama yagona echimga ega bo'ladi.
Teorema. Agar (a; m)=d bo'lib, b son d ga bo'linmasa, u holda (3) taqqoslama echimga ega emas.
Teorema. Agar (3) taqqoslamada (a; m)=d bo'lib, b son d ga bo'linsa, u holda (3) taqqoslama soni d ga teng bo'lgan ushbu
— m (d - 1)m
a, an--.....an--
dd
(5)
a b, . m.
echimlarga ega bo'lib, bundagi a echim — X = — (mod —) taqqoslamaning yagona
dad
echimi bo'ladi.
Ta'rif. Agar f(x) = aoxp+aixn-1 +...+an-i x+an ,a*eZ, r-tub son, aocon r ga bo'linmasa, u holda ushbu
f(x) = 0(mod p) (6)
taqqoslamaga tub modulli p-darajali bir nomat'lumli taqqoslama deyiladi. Teorema. Agar (6) taqqoslamada ao bosh koeffitsient r ga bo'linmasa, u holda (6) taqqoslama bosh koeffitsienti 1 ga teng bo'lgan boshqa bir taqqoslamaga teng kuchli bo'ladi.
Teorema. Agar f(x) va g(x) koeffitsientlari butun sonlardan iborat ko'phadlar bo'lsa, u
holda
f(x) = 0(mod p), (7)
f(x)-(xp-x)g(x) = 0(modp) (8)
taqqoslamalar teng kuchli bo'ladi.
Teorema. Darajasi n (n>r) bo'lgan r tub modulli taqqoslama darajasi r-1 dan katta bo'lmagan taqqoslamaga teng kuchli bo'ladi.
Teorema. Tub modulli n-darajali taqqoslama echimlari soni n tadan ortiq emas.
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