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.
The set of primes: {2, 3, 5, . . .}.
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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.
The set of primes: {2, 3, 5, . . .}.
An integer greater than 1 that is not a prime is termed composite.
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Motivation to learn about prime numbers
• public key cryptography
• hashing
• pseudorandom number generators
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Few elementary properties of primes
• if p is a prime and p|ab, then p|a or p|b
* if p 6 |a then gcd(p, a) = 1; apply 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 together with the help of above
• 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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Prime factor of a composite integer
If n is a composite integer, then n has a prime factor not exceeding
√
n.
√
For a positive integer n > 2, let S be the set of primes less than n. Let S0 be
the set comprising of every integer in [n/2, n] that is not divisible by any
integer in S. Then the set of primes less than n is precisely equal to S ∪ S0 .
(This is the basis for the sieve of Eratosthenes.)
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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
* if m is a divisor of Fk and Fn (for k < n) then m|2; but since the Fermat numbers are odd integers,
then m must be odd; hence m = 1
* i.e., 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
n−1
• If pn is the nth prime number, then pn ≤ 22
* induction on n while noting that pn+1 ≤ p1 p2 . . . pn + 1
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Bound on the nth prime in terms of n
n−1
• If pn is the nth prime number, then pn ≤ 22
* 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, . . . ,.
• The 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)
— neither of these are 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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