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Presentation on the topic "Number Theory" for Continuous Assessment, Slides of Discrete Mathematics

Maulana Abul Kalam Azad University of TechnologyDiscrete Mathematics

An introduction to number theory in discrete mathematics. It covers definitions, the fundamental theorem of arithmetic, modular arithmetic, infinitude of primes, Diophantine equations, Fermat’s Little Theorem, and the Chinese Remainder Theorem. Number theory is an essential part of discrete mathematics and has applications in computer science, cryptography, and other fields. a proof of Euclid's theorem that there are infinitely many primes and explains how Diophantine equations can be used to solve problems in number theory.

Typology: Slides

2022/2023

Available from 03/14/2023

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Section summary
Introduction to Number Theory in Discrete Mathematics
Definitions
The Fundamental Theorem of Arithmetic &
Modular Arithmetic
Infinitude of Primes
Diophantine Equations
Fermat’s Little Theorem
The Chinese Remainder Theorem
Conclusion
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Download Presentation on the topic "Number Theory" for Continuous Assessment and more Slides Discrete Mathematics in PDF only on Docsity!

Section summary

 Introduction to Number Theory in Discrete Mathematics

 Definitions

 The Fundamental Theorem of Arithmetic &

Modular Arithmetic

 Infinitude of Primes

 Diophantine Equations

 Fermat’s Little Theorem

 The Chinese Remainder Theorem

 Conclusion

Introduction to Number Theory in Discrete

Mathematics

 Number theory is the branch of mathematics that deals with the properties and

relationships of integers. It is an essential part of discrete mathematics, which is the

study of mathematical structures that are discrete rather than continuous.

 Number theory has a wide range of applications in computer science, cryptography,

and other fields. It is an important part of the foundation of mathematics and is used

to solve problems in many areas.

The Fundamental Theorem of Arithmetic &

Modular Arithmetic

Theorem: Every positive integer greater

than 1 can be written uniquely as a prime

or as the product of two or more primes

where the prime factors are written in

order of nondecreasing size.

Modular arithmetic is a type of
arithmetic in which numbers wrap
around after a certain number. It is
an important part of number theory
and has many applications in
cryptography and computer science

Infinitude of Primes

Euclid(325 B.C.E. – 265 B.C.E.) Theorem: There are infinitely many primes. (Euclid) Proof: Assume finitely many primes: p1, p2, ….., pn Let q = p1p2∙∙∙ pn + 1 Either q is prime or by the fundamental theorem of arithmetic it is a product of primes. But none of the primes pj divides q since if pj | q, then pj divides q − p1p2∙∙∙ pn = 1. Hence, there is a prime not on the list p1, p2, ….., pn. It is either q, or if q is composite, it is a prime factor of q. This contradicts the assumption that p1, p2, ….., pn are all the primes. Consequently, there are infinitely many primes. This proof was given by Euclid The Elements. The proof is considered to be one of the most beautiful in all mathematics.

Fermat’s Little Theorem

 Fermat’s Little Theorem: If p is prime and a is an integer not divisible by p,

then ap- 1 ≡ 1 (mod p). Furthermore, for every integer a , we have ap ≡ a

(mod p).

Fermat’s little theorem is useful in computing the remainder modulo p of large

powers of integers.

Example: Find 7222 mod 11.By Fermat’s little theorem, we know that 710 ≡ 1

(mod 11), and so (710 )k ≡ 1 (mod 11), for every positive integer k.

Therefore,7222 = 722∙ = (710)2272 ≡ (1)22 ∙ 49 ≡ 5 (mod 11).Hence, 7222 mod 11

The Chinese Remainder Theorem

To construct a solution first let Mk=m/mk for k = 1,2,…,n and m = m1m2 ∙ ∙ ∙ mn. Since gcd(mk ,Mk ) = 1, by Theorem 1, there is an integer yk , an inverse of Mk modulo mk, such that Mk yk ≡ 1 ( mod mk ). Form the sum x = a1 M1 y1 + a2 M2 y2 + ∙ ∙ ∙ + an Mn yn. Note that because Mj ≡ 0 ( mod mk) whenever j ≠k , all terms except the kth term in this sum are congruent to 0 modulo mk. Because Mk yk ≡ 1 ( mod mk ), we see that x ≡ ak Mk yk ≡ ak( mod mk), for k = 1,2,…,n. Hence, x is a simultaneous solution to the n congruences. x ≡ a1 ( mod m1) x ≡ a2 ( mod m2) x ≡ an ( mod mn)

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