T A D Q I Q O T L A R
jahon ilmiy – metodik jurnali
https://scientific-jl.com
61-son_3-to’plam_May-2025
38
ISSN:3030-3613
EYLER VA FERMA TEOREMASI
Zaxiriddinova Shahlo Zaxiriddin qizi
Shahrisabz davlat pedagogika instituti
Matematika va ta’limda axborot
texnologiyasi kafedrasi o’qituvchisi
Axmedova Yulduz Baxodirovna
Shahrisabz davlat pedagogika
instituti ,,Matematika va informatika’’
yo’nalishi 2-bosqich talabasi
Anotatsiya
Ushbu maqolada sonlar nazariyasining eng muhim teoremalari — Ferma kichik
teoremasi va Eller teoremasining nazariy asoslari va amaliy ahamiyati keng yoritilgan.
Har ikki teorema modulli arifmetikaning muhim qismini tashkil qiladi va tub sonlar
bilan bog‘liq ko‘plab masalalarni soddalashtirishda muhim vosita bo‘lib xizmat qiladi.
Maqolada avvalo har bir teoremaning mazmuni, matematik ifodalanishi va isboti
tushuntirilgan. Keyin esa ularning o‘zaro bog‘liqligi, farqlari va umumlashtirilgan
shakllari ko‘rib chiqilgan.
Shuningdek, maqolada Ferma va Eller teoremalari asosida modulli tenglamalar
va sonlar ustida bajariladigan arifmetik amallarni soddalashtirish yo‘llari amaliy
misollar bilan izohlangan. Ayniqsa, ushbu teoremalarning kriptografiya sohasida,
xususan RSA algoritmida qanday qo‘llanilishi keng yoritilgan. Maqola ushbu
matematik nazariy bilimlarning raqamli xavfsizlik, axborot texnologiyalari va
zamonaviy algoritmik tizimlar uchun qanday ahamiyat kasb etishini ko‘rsatadi. Bu
orqali talabalar, o‘qituvchilar va tadqiqotchilar sonlar nazariyasini chuqurroq
o‘rganishlari hamda uni amaliyotda qo‘llash imkoniyatiga ega bo‘ladilar.
Annotation
This article presents a comprehensive exploration of two key theorems in
number theory: Fermat’s Little Theorem and Euler’s Theorem. Both theorems are
essential tools in modular arithmetic and play a significant role in simplifying complex
operations involving prime numbers and integer relations. The article begins by
explaining the core concepts, mathematical formulations, and proofs of each theorem.
Their connections, differences, and generalizations are then discussed in detail.
Furthermore, the article examines practical applications of these theorems
through illustrative problem-solving examples involving modular equations and
arithmetic operations. Special attention is given to their role in the field of
cryptography, particularly within the RSA encryption algorithm, which forms the
T A D Q I Q O T L A R
jahon ilmiy – metodik jurnali
https://scientific-jl.com
61-son_3-to’plam_May-2025
39
ISSN:3030-3613
backbone of modern digital security. The paper highlights how these theoretical
concepts extend far beyond pure mathematics and are actively used in real-world
applications such as secure communication and data protection. The article serves as a
valuable resource for students, educators, and researchers aiming to deepen their
understanding of number theory and its practical utility.
Kalit so‘zlar:
Ferma kichik teoremasi, Eller teoremasi, sonlar nazariyasi,
modulli arifmetika, tub son, kriptografiya, RSA algoritmi, matematik isbot, modulli
tenglama, raqamli xavfsizlik
Keywords:
Fermat's Little Theorem, Euler’s Theorem, number theory, modular
arithmetic, prime number, cryptography, RSA algorithm, mathematical proof, modular
equation, digital security
Kirish
Matematikaning eng qadimiy va chuqur yo‘nalishlaridan biri bo‘lgan sonlar
nazariyasi — butun sonlar xossalarini o‘rganishga yo‘naltirilgan bo‘lib, u zamonaviy
matematik tarmoqlar, ayniqsa kriptografiya, raqamli xavfsizlik, va kompyuter fanlari
asoslarini tashkil etadi. Sonlar nazariyasida muhim o‘rin egallagan teoremalardan
ikkitasi — bu Ferma kichik teoremasi va Eller teoremasidir. Ular modulli
arifmetikaning asosiy tamoyillarini belgilab beradi va tub sonlar bilan ishlashda
matematik amallarni soddalashtirishda keng qo‘llaniladi.
Ferma kichik teoremasi XVII asrda Pyer Ferma tomonidan ilgari surilgan bo‘lib,
u orqali har qanday tub son bilan ishlovchi modulli tenglamalarni samarali tarzda
yechish mumkin. Bu teorema sonlar ustida bajariladigan murakkab arifmetik amallarni
yengillashtiradi va matematik mantiqni chuqur tushunishga xizmat qiladi. Eller
teoremasi esa Ferma teoremasining umumlashtirilgan shakli bo‘lib, u faqat tub sonlar
emas, balki har qanday musbat butun sonlar bilan o‘zaro tub bo‘lgan sonlar orasidagi
munosabatni modulli shaklda ifodalash imkonini beradi. Bu esa modulli
arifmetikaning qo‘llanish doirasini yanada kengaytiradi.
Ushbu teoremalar zamonaviy texnologiyalar, ayniqsa axborot xavfsizligi
sohasida muhim ahamiyat kasb etadi. Aynan ular asosida qurilgan RSA shifrlash
algoritmi internetdagi ko‘plab maxfiy muloqot va ma’lumot almashuvni himoyalashda
foydalaniladi. Shunday ekan, Ferma va Eller teoremalari nafaqat nazariy, balki amaliy
jihatdan ham nihoyatda muhim hisoblanadi.
Ushbu maqolada aynan shu teoremalar chuqur tahlil qilinadi, ularning
matematik mazmuni, formulalari, isbotlari va amaliy misollardagi qo‘llanilish usullari
ko‘rib chiqiladi. Shuningdek, ular yordamida qanday qilib modulli tenglamalar
yechilishi va katta sonlar ustida tez hisob-kitoblar olib borilishi mumkinligi izohlanadi.
Bu nafaqat matematikani chuqurroq anglashga yordam beradi, balki ularni real
hayotda, texnologiyada qo‘llash imkoniyatini ham ochib beradi.
T A D Q I Q O T L A R
jahon ilmiy – metodik jurnali
https://scientific-jl.com
61-son_3-to’plam_May-2025
40
ISSN:3030-3613
Introduction
Number theory, one of the oldest and most profound branches of mathematics,
is primarily focused on the properties and relationships of integers. It forms the
theoretical foundation for various modern mathematical disciplines, particularly
cryptography, digital security, and computer science. Among the many theorems that
play a crucial role in number theory, two stand out for their fundamental significance
— Fermat’s Little Theorem and Euler’s Theorem. These theorems establish key
principles of modular arithmetic and are widely used to simplify operations involving
prime numbers.
Fermat’s Little Theorem, proposed in the 17th century by Pierre de Fermat,
provides an efficient way to solve modular equations involving prime numbers. It
allows complex arithmetic operations to be reduced in modular systems and enhances
understanding of mathematical logic. Euler’s Theorem, considered a generalization of
Fermat’s, extends these principles to any pair of relatively prime positive integers,
making it applicable in a broader range of problems within modular arithmetic.
Both theorems have immense importance not only in theoretical mathematics but also
in practical applications, especially in information security. They form the
mathematical basis for the RSA encryption algorithm, which is one of the most widely
used systems to ensure secure communication over the internet. As such, understanding
these theorems is crucial for both mathematicians and computer scientists alike.
This article aims to explore these two theorems in depth, examining their mathematical
structure, proofs, and practical use through real examples. It also highlights how they
enable efficient computation with large numbers and the solution of modular equations
— serving not only to deepen one’s mathematical insight but also to connect theory
with real-world technological applications.
Asosiy qism
Ferma kichik teoremasi va Eller teoremasi sonlar nazariyasining eng muhim
teoremalari qatoriga kiradi. Ular modulli arifmetika, tub sonlar va algoritmlarda keng
qo‘llaniladi. Quyida har bir teorema alohida ko‘rib chiqilib, ularning o‘ziga xosliklari,
formulalari, isbotlari va amaliy qo‘llanilishi haqida batafsil ma’lumot beriladi.
Ferma kichik teoremasi
Ferma va Eller teoremalari o‘rtasidagi bog‘liqlik
Ferma teoremasi — maxsus holat, ya’ni tub son bo‘lganidagi holat, Eller teoremasi
esa umumiyroq, ya’ni har qanday musbat butun son bo‘lishi mumkin. Har ikki teorema
modulli arifmetika asosida ishlaydi va bir-birini to‘ldiradi.
Amaliy qo‘llanishi
Ferma va Eller teoremalari zamonaviy kriptografik algoritmlar asosida qo‘llaniladi.
Ayniqsa, RSA algoritmida ochiq kalitni yaratish va yopiq kalitni topish uchun aynan
ushbu teoremalardan foydalaniladi.
T A D Q I Q O T L A R
jahon ilmiy – metodik jurnali
https://scientific-jl.com
61-son_3-to’plam_May-2025
41
ISSN:3030-3613
RSA algoritmida:
Ikki katta tub son va tanlanadi.
, va hisoblanadi.
Shunda maxfiy kalitni topishda dan foydalaniladi.
Shuningdek, bu teoremalar modulli darajali hisoblash, raqamli imzolar, ma’lumotlarni
shifrlash va raqamli xavfsizlikni ta’minlashda keng qo‘llaniladi
Yechiladigan masalalar turlari
Ferma va Eller teoremalari asosida quyidagi turdagi masalalar yechiladi:
1. Modulli darajalar hisoblash:
2. O‘zaro tub sonlar aniqlash
3. Raqamli shifrlash va deshifrlash masalalari
4. Modular tenglamalar yechimi
5. Kriptografik kalitlarni aniqlash
Main part
Farm small theorem and Eller's theorem are numbers
among the most important theorems of his theory.
They are wide in modular arithmetic, prime numbers and algorithms
applied. Below, each theorem is considered separately,
their identities, formulas, proofs and practical
details of the application will be given.
Farm small theorem
Relation between farm and Eller theorems
The farm theorem is a special case, i.e. one that is prime
the case, while Eller's theorem is more general, i.e. any
can be a positive integer. Both theorems are modular
it works on the basis of arithmetic and complements each other.
Practical application
Farm and Eller theorems-modern cryptographic algorithms
applied on the basis. Especially open in the RSA algorithm
this is exactly what you need to create a key and find a closed key
theorems are used.
In the RSA algorithm:
Two large prime numbers and are selected.
, and is.
Then when finding the secret key, dan is used.
Also, these theorems are modular level computing, numerical
signatures, data encryption and digital security
widely used in supply
T A D Q I Q O T L A R
jahon ilmiy – metodik jurnali
https://scientific-jl.com
61-son_3-to’plam_May-2025
42
ISSN:3030-3613
Types of issues to be solved
Based on the farm and Eller theorems, the following types of issues are
undo:
1. Modular levels calculation:
2. Determination of mutual prime numbers
3. Digital encryption and decryption issues
4. Modular equation solutions
5. Cryptographic key recognition
Foydalanilgan adabiyotlar
1.
Turaev D. T. – Diskret matematika va matematik mantiq. Toshkent: “Fan va
texnologiya”, 2015.
2.
Rasulov A. A. – Sonlar nazariyasiga kirish. Toshkent: O‘zbekiston Milliy
Universiteti, 2007.
3.
Kadyrov B. va boshqalar – Kriptografiya asoslari. Toshkent: Iqtisodiyot, 2018.
4.
https://en.wikipedia.org/wiki/Fermat%27s_little_theorem
5.
https://en.wikipedia.org/wiki/Euler%27s_theorem_(mathematics)
6.
Rosen K. H. – Discrete Mathematics and Its Applications, 7th Edition, McGraw-
Hill, 2012.
References
1.
Turaev D. T. – Discrete Mathematics and Mathematical Logic. Tashkent: “Fan va
texnologiya”, 2015.
2.
Rasulov A. A. – Introduction to Number Theory. Tashkent: National University of
Uzbekistan, 2007.
3.
Kadyrov B. et al. – Foundations of Cryptography. Tashkent: Iqtisodiyot, 2018.
4.
https://en.wikipedia.org/wiki/Fermat%27s_little_theorem
5.
https://en.wikipedia.org/wiki/Euler%27s_theorem_(mathematics)
6.
Rosen K. H. – Discrete Mathematics and Its Applications, 7th Edition, McGraw-
Hill, 2012.
