Prime and Relatively Prime Numbers

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Prime and Relatively Prime Numbers

• Divisors We say that b ? 0 divides a if a mb
  for some m, where a, b and m are integers.
• b divides a if there is no remainder on division.
• The notation ba is commonly used to mean that b
  divides a.
• If ba, we say that b is a divisor of a.

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Prime and Relatively Prime Numbers (contd)

• If a1, then a ? 1.
• If ab and ba, then a ? b.
• Any b ? 0 divides 0.
• If bg and bh, then b(mg nh) for arbitrary
  integers m and n.

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Prime and Relatively Prime Numbers (contd)
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Prime and Relatively Prime Numbers (contd)
Table 7.1 Primes under 2000
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Prime and Relatively Prime Numbers (contd)

• The above statement is referred to as the prime
  number theorem, which was proven in 1896 by
  Hadaward and Poussin.

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Prime and Relatively Prime Numbers (contd)
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Prime and Relatively Prime Numbers (contd)

• Whether there exists a simple formula to generate
  prime numbers?
• An ancient Chinese mathematician conjectured that
  if n divides 2n - 2 then n is prime. For n 3, 3
  divides 6 and n is prime. However, For n 341
  11 ? 31, n dives 2341 - 2.
• Mersenne suggested that if p is prime then Mp
  2p - 1 is prime. This type of primes are referred
  to as Mersenne primes. Unfortunately, for p 11,
  M11 211 -1 2047 23 ? 89.

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Prime and Relatively Prime Numbers (contd)

• Fermat conjectured that if Fn 22n 1, where n
  is a non-negative integer, then Fn is prime. When
  n is less than or equal to 4, F0 3, F1 5, F2
  17, F3 257 and F4 65537 are all primes.
  However, F5 4294967297 641 ? 6700417 is not a
  prime bumber.
• n2 - 79n 1601 is valid only for n lt 80.
• There are an infinite number of primes of the
  form 4n 1 or 4n 3.
• There is no simple way so far to gererate prime
  numbers.

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Prime and Relatively Prime Numbers (contd)

• Factorization of an integer as a product of prime
  numbers
• Example 91 7 ? 13 11011 7 ? 112 ? 13.
• Useful for checking divisibility and relative
  primality to be discussed later.
• Factorization is in gereral difficult.

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Prime and Relatively Prime Numbers (contd)

• Define notation gcd(a,b) to mean the greatest
  common divisor of a and b.
• The positive integer c is said to be the gcd of a
  and b if
• ca and cb
• any divisor of a and b is a dividor of c.
• Equivalently, gcd(a,b) maxk, such that ka and
  kb
• gcd(a,b) gcd(-a,b) gcd(a,-b) gcd(-a,-b)
  gcd(a,b)

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Prime and Relatively Prime Numbers (contd)

• gcd(a,0) a.
• Factorization is one possible but in general
  inefficient way to calculate gcd. Whereas,
  Euclids algorithm (to be discussed later) is
  more efficient.
• Relative primality
• the integers a and b are relatively prime if they
  have no prime factors in common
• or equivalently, their only common factor is 1
• or equivalently, gcd(a,b) 1

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Modular Arithmetic
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Modular Arithmetic (contd)

• Examples
• a 11 n 7 11 1 ? 7 4 r 4.
• a -11 n 7 -11 (-2) ? 7 3 r 3.
• If a is an integer and n is a positive integer,
  define a mod n to be the remainder when a is
  divided by n.
• Then, a ?a/n? ? n (a mod n)
• Example 11 mod 7 4 -11 mod 7 3.

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Modular Arithmetic (contd)
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Modular Arithmetic (contd)

• Properties of modular arithmetic operations
• Proof of Property 1
• Define (a mod n) ra and (b mod n) rb. Then a
  ra jn and b rb kn for some integers j and
  k. Then,
• (ab) mod n (ra jn rb kn) mod n
• (ra rb (j k)n)
  mod n
• (ra rb) mod n
• (a mod n) (b mod n)
  mod n

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Modular Arithmetic (contd)
? Examples for the above three properties
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Modular Arithmetic (contd)

• Properties of modular arithmetic
• Let Zn 0,1,2,,(n-1) be the set of residues
  modulo n.

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Modular Arithmetic (contd)

• Properties of modular arithmetic (contd)
• if (a b) ? (a c) mod n, then b ? c mod n (due
  to the existence of an additive inverse)
• if (a ? b) ? (a ? c) mod n, then b ? c mod n
  (only if a is relatively prime to n due to the
  possible absence of a multiplicative inverse)
• e.g. 6 ? 3 18 ? 2 mod 8 and
• 6 ? 7 42 ? 2 mod 8 but
• 3 ? 7 mod 8 (6 is not relatively prime
  to 8)
• If n is prime then the property of multiplicative
  inverse holds (from a ring to a field).

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Modular Arithmetic (contd)

• Properties of modular arithmetic (contd)

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Fermats and Eulers Theorems

• Fermats
• theorem

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Fermats and Eulers Theorems (contd)

• Fermats theorem (contd)
• alternative form
• if p is prime and a is any positive integer,
  then
• ap ? a mod p
• example p 5, a 3, 35 243 ? 3 mod 5

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Fermats and Eulers Theorems (contd)

• Eulers totient function

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Fermats and Eulers Theorems (contd)
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Fermats and Eulers Theorems (contd)

• Eulers totient function (contd)
• if n is the product of two primes p and q
• f(n) pq (q 1)(p 1) 1
• pq (p q) 1
• (p 1) ? (q 1)
• f (p) ? f (q)

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Fermats and Eulers Theorems (contd)

• Eulers theorem

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Fermats and Eulers Theorems (contd)

• Eulers totient function (contd)

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Testing for Primality

• If p is an odd prime, then the equation
• x2 ? 1 (mod p) has only two solutions, 1 and
  -1.

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Testing for Primality (contd)
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Testing for Primality (contd)

• Probabilistic primality test

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Euclids Algorithm
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Euclids Algorithm (contd)
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Euclids Algorithm (contd)
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Euclids Algorithm (contd)
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Extended Euclids Algorithm
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Chinese Remainder Theorem
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Chinese Remainder Theorem (contd)
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Discrete Logarithms
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Discrete Logarithms (contd)