Number Theory
Prime Numbers
R. Inkulu
http://www.iitg.ac.in/rinkulu/
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Outline
1
Introduction
2
Infinitude of primes
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Prime Number: definition
An integer p > 1 is called a prime number, or simply a prime, if its only
positive divisors are 1 and p. An integer greater than 1 that is not a prime is
termed composite.
The set of primes: {2, 3, 5, . . .}.
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Motivation to learn about prime numbers
• public key cryptography
• hashing
• pseudorandom number generators
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Relatively prime: definitions
• Two integers a and b, not both of which are 0, are said to be relatively
prime (a.k.a. coprime) whenever gcd(a, b) = 1.
• The integers a1 , a2 , . . . , an are mutually relatively prime if
gcd(a1 , a2 , . . . , an ) = 1.
• The integers a1 , a2 , . . . , an are pairwise relatively prime if, for each pair
of integers ai and aj , gcd(ai , aj ) = 1.
Note that pairwise relatively prime integers must be mutually relatively prime,
but the converse is not necessarily true.
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Few elementary properties of primes
• if p is a prime and p|ab, then p|a or p|b
from Euclid’s lemma
• if p is a prime and p|a1 a2 . . . an , then p|ak for some k, where 1 ≤ k ≤ n
induction on n
• if p, q1 , q2 , . . . , qn are all primes, and p|q1 q2 . . . , qn , then p = qk for
some k, where 1 ≤ k ≤ n
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Fundamental Theorem of Arithmatic
Every positive integer n > 1 can be written uniquely as a prime or as the
product of two or more primes where the prime factors are listed in sorted
order.
• induction on n to show that n is the product of two or more primes
• proof by contradiction to show the uniqueness
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Unique factorization theorem: a corollary to
fundamental theorem of arithmatic
Any positive integer n > 1 can be written uniquely in a canonical form
n = pk11 pk22 . . . pkr r where for i = 1, 2, . . . , r, each ki is a positive integer and
each pi is a prime, with p1 < p2 < . . . < pr .
Ex. 360 = 23 ∗ 32 ∗ 5
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Prime factor of a composite integer
If n is a composite integer, then n has a prime factor not exceeding
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√
n.
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Legendre’s theorem
The number n! contains the prime factor p exactly
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P
n
k≥1 ⌊ pk ⌋
times.
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Outline
1
Introduction
2
Infinitude of primes
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Euclid’s proof
Euclid’s Theorem: There are infinitely many prime numbers.
let S = {p1 , p2 , . . . , pn } be the finite set of primes
• reaching a contradiction: r = p1 p2 . . . pn + 1 is either a prime not listed
in S or has a prime factor 1 < p < r not listed in S
(proof strategy:
nonconstructive existence)
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Christian Goldbach’s proof of Euclid’s theorem
n
Consider Fermat numbers Fern = 22 + 1 for n = 0, 1, 2, . . . 1
• Πn−1
k=0 Ferk = Fern − 2
proof by induction on n
• since the Fermat numbers are odd integers, gcd of any two of them must
be odd
• any two Fermat numbers are relative prime; hence, proved
1
initially, Fermat conjectured that all the Fermat numbers are primes, which is found to be
false for n = 5
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Bound on the nth prime in terms of n
• If pn is the nth prime number, then pn ≤ 22
n−1
induction on n while noting that pn+1 ≤ p1 p2 . . . pn + 1
n
• Corollary: For n ≥ 1, there are at least n + 1 primes less than 22 .
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Popular theorems
• Dirichlet’s theorem: For any relatively prime positive integers a and b,
there are infinitely many primes of the form an + b, for n = 1, 2, . . . ,.
• Lagrange’s prime number theorem: Let x be a positive real number and
let the function π(x) denote the number of primes not exceeding x, then
limx→∞ ( π(x)
x
) = 1.
lge (x)
— both are not proved in class
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Popular conjectures
• Twin Prime Conjecture: there are infinitely many pairs of primes p and
p+2
• Goldbach’s Conjecture: every even positive integer greater than 2 can be
written as the sum of two primes
• The n2 + 1 Conjecture: there are infinitely many primes of the form
n2 + 1, where n is a positive integer
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