Manindra Agrawal

1
A Brief History of Primes

• Manindra Agrawal
• Dept of Computer Science
• Indian Institute of Technology
• Kanpur

2
What Are Primes?

• Number n is prime if it is greater than 1 and
  is divisible only by 1 and itself. Otherwise, it
  is composite.
• Examples 2, 3, 5, 7 are first four primes
  while 4 2 2, 6 2 3 are composites.

3
Why primes?

• Fundamental Theorem of Arithmetic Every
  number can be written as a product of prime
  numbers in a unique way.
• Primes are building blocks for all numbers.
• Understanding them is crucial to understanding
  properties of numbers.

4
Where are they?

• Here is the list of primes amongst first 100
  numbers
• 2, 3, 5, 7,
• 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,
  53, 59, 61, 67, 71, 73,
• 79, 83, 89, 93

5
Some Observations

• There are 25 primes less than 100 (25).
• Of these,
• 4 are less than 10 (40).
• 15 are less than 50 (30).
• 21 are less than 75 (28).
• The fraction of primes is decreasing as the
  interval size increases.

6
Some Questions

• How many primes exist?
• Can we estimate the number of primes less than
  some given number n?
• How do we find out if number n is prime?
• How do we generate all primes less than some
  given number n?
• We will go through the history of these
  questions.

7
300 BC Euclids Elements

• Proved the Fundamental Theorem of Arithmetic.
• Proved that there exist infinitely many primes.
• This answers Question 1!

8
230 BC The Sieve of Eratosthenes

• Eratosthenes gave a method to generate all prime
  numbers between 1 and n
• Write down all numbers from 2 to n.
• Take the first uncrossed number, say p and cross
  all multiples of p except p itself.
• Repeat 2 until no number can be crossed out.

9
The Sieve at Work
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
10
The Sieve at Work
2
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43
45
47
49
51
53
55
57
59
61
63
65
67
69
71
73
75
77
79
81
83
85
87
89
91
93
95
97
99
11
The Sieve at Work
2
3
5
7
11
13
17
19
23
25
29
31
35
37
41
43
47
49
53
55
59
61
65
67
71
73
77
79
83
85
89
91
95
97
12
The Sieve at Work
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
49
53
59
61
67
71
73
77
79
83
89
91
97
13
The Sieve at Work
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
49
53
59
61
67
71
73
79
83
89
97
14
The Sieve at Work
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
49
53
59
61
67
71
73
79
83
89
97
15
Question 4 is Also Answered!

• We can use the same method to answer Question 3
  as well
• to test if number n is prime, generate all primes
  up to n and see if n is one of them.
• But this is an overkill we just need to find if
  n is prime!
• There should be a faster method for testing if n
  is prime.

16
Yet to be Answered

• Question 2 How many primes up to n?
• Question 3 How to quickly test if n is prime?
• Answering these should not take long

17
200 BC 1600 AD The Dark Ages
Almost nothing of significance happened for primes
18
1601-65 Pierre de Fermat and his Little Theorem

• Made several contributions to the theory of
  primes.
• The most important one is
• Fermats Little Theorem For any prime n and
  any number a, an (mod n) a.
• an multiply a to itself n-1 times
• x (mod n) divide x by n and take the remainder

19
The Magic of Fermats Little Theorem

• 25 (mod 5) 32 (mod 5) 6 5 2 (mod 5) 2
• 35 (mod 5) 243 (mod 5) 48 5 3 (mod 5) 3
• 211 (mod 11) 2048 (mod 11) 186 11 2 (mod
  11) 2
• 47 (mod 7) 16384 (mod 7) 2340 7 4 (mod 7)
  4

20
FLT as Primality Test?

• Can we use this to test if n is prime?
• Test Given n, choose some a, compute an (mod n)
  and check if it equals a.
• Ancient Chinese ( 400 BC) believed that the
  above test for a 2 will correctly determine
  primes.

21
Is it Really Quick?

• Problems
• Multiplying a with itself n-1 times will take a
  long time when n is large.
• And the numbers will grow very big!
• Solution First compute a2 (mod n), then multiply
  it with itself to compute a4 (mod n), then a8
  (mod n) etc!

22
FLT Based Test is Wrong!

• Much later, Korsalt (1899) and Carmichael (1901)
  noted that some composites also satisfy this
  test!
• Such numbers are called Carmichael numbers.
• The smallest Carmichael number is 561 3 11
  17.
• Recently (1994) it was shown that there are
  infinitely many Carmichael numbers.

23
Formula for Primes?

• Prime numbers appear chaotically distributed on
  the number line.
• Is there a formula that describes primes?
• This would also (hopefully) solve Question 3
• One can perhaps quickly determine if a number
  satisfies the formula.

24
Fermat Numbers

• The problem turned out to be very difficult.
• Fermat suggested a formula for a subset of
  primes
• 22m 1 is always prime
• Nearly 100 years later, Euler showed that for m
  5 the formula gives composite number.

25
Mersenne Primes

• Marin Mersenne (1588-1648) studied primes of the
  form 2m 1 where m itself is prime.
• These are called Mersenne primes.
• Not all such numbers are primes though.
• Example m 11 gives 211 1 2047 23 89.
• The largest known prime today is a Mersenne
  prime 213466917-1 (nearly 4 million digits).

26
Eulers Formula

• Euler (1707-83) came up with the following
  formula
• m2 m 41
• The first 40 values of this formula are primes!

27
Can Question 2 be Answered Positively?

• Primes do not seem to follow any pattern.
• Perhaps there is no good estimate of their number
  up to n?

28
1808 Perhaps Yes!

• Legendre (1752-1833) and Gauss (1777-1855)
  observed that
• Number of primes less than n converges to
  n/log n as n increases!
• log n ? number of digits in n

29
1859 Riemanns Zeta Function

• Riemann (1826-66), a student of Gauss, defined
  the following function
• where y is a complex number.

30
Zeta Function And Primes

• Zeta function has deep connection with primes.
• Understanding it well is key to understanding
  distribution of primes.
• The quest for understanding zeta function led to
  the development of Complex Analysis.

31
Riemanns Hypothesis

• Riemann conjectured that all interesting zeroes
  of zeta function lie on a straight line on
  complex plane.
• If proven, it will answer a number of questions
  about distribution of primes including Question
  2.
• It remains open till date!

32
1896 Question 2 is Answered!

• Riemann along with Chebyshev (1821-94) made
  considerable progress towards answering the
  question.
• Hadamard (1865-1963) and Valee Poussin
  (1866-1962) finally proved that the number of
  primes less than n indeed converges to n / log n.

33
Back to Formulae for Primes

• Many interesting formulae for primes were given
  over a period of time.
• For example, Willans in 1964 gave
• P(n) sin2(p(n-1)!2/n) / sin2(p/n)
• P(n) is 1 if n is prime, 0 otherwise.

34
Is This A Good Formula?

• No, because it gives no new insight into prime
  numbers.
• It is directly based on the property of primes
  that n is prime if and only if (n-1)! 1 is
  divisible by n.
• It is also not fast
• One needs to multiply lots of times to compute
  (n-1)! 1.

35
Formalizing Notion of Fast Method

• Notion of computation developed by Turing, Church
  etc in 1930s gives a precise way to define a fast
  method
• Count the number of operations performed by a
  method.
• A method is fast if it takes at most a polynomial
  in log n number of steps.
• In other words, a fast method takes at most (log
  n)c steps for constant c.

36
Methods Seen So Far Are Inefficient

• Sieve of Eratosthenes crosses or checks all
  numbers less than n.
• So it takes at least n steps.
• Willans formula requires computation of (n-1)!.
• This needs n-2 multiplication.

37
1972 The First Efficient Method

• Gary Miller used Fermats Little Theorem to
  design an efficient method.
• It takes roughly (log n)4 steps.
• However, he could prove correctness of his method
  only after assuming a form of Riemann Hypothesis.

38
1974-76 Two More Efficient Methods

• Solovay and Strassen designed an efficient method
  that is unconditionally correct.
• However, their method makes mistakes sometimes!
• When n is composite, then the method has a small
  chance of deciding incorrectly.
• Rabin modified Millers algorithm to design a
  method with similar properties.

39
1976-77 Public-Key Cryptosystems

• Cryptosystems are used to code secret information
  while transmitting.
• Diffie and Hellman proposed the notion of
  public-key cryptosystems.
• Using this, even two strangers can exchange
  secret information.
• Next year, Rivest, Shamir, and Adleman devised
  the first such system.
• This is the well-known RSA system.

40
RSA Needs Large Primes!

• For ensuring secrecy, RSA system needs to use
  very large prime numbers.
• These days, 160 digit prime numbers are routinely
  used in RSA.
• This made efficient method for testing if a
  number is prime very useful!

41
1983 Another (almost) Efficient Method

• Adleman, Pomerance and Rumeli designed a test
  that is
• Unconditionally correct
• Never makes a mistake
• However, it is not efficient, although is very
  close
• It needs (log n)c loglog n operations.

42
August 2002An Efficient Method Without Drawbacks

• A few months ago, Agrawal, Kayal, and Saxena
  designed a method that is
• Efficient
• Unconditionally correct
• Never makes a mistake
• Question 3 is thus completely answered.

43
Future Unanswered Questions

• Despite impressive strides in understanding
  primes, they are still not well understood.
• Many questions about them are unanswered.

44
Riemann Hypothesis

• Perhaps the most important question in
  mathematics.
• A positive answer will greatly increase our
  knowledge of primes.
• There is a 1 million prize for solving this!

45
Goldbachs Conjecture

• This conjecture states that every even number is
  a sum of two prime numbers.
• Example 16 5 11, 24 11 13,
• Proposed by Goldbach in 1742.

46
Twin Prime Conjecture

• Twin primes are pairs n, n2 such that both are
  primes.
• Example 3-5 11-13 17-19 71-73
• The conjecture states that there exist infinitely
  many twin primes.

47
How Far Are Two Consecutive Primes?

• The conjecture states that if n is prime then the
  next prime is less than n c(log n)3 for some
  constant c.
• The best result known is that next prime is less
  than 2n.

48
Factoring Numbers Efficiently

• Design efficient method for decomposing any
  number into its prime factors.
• No efficient algorithm is known.
• The security of RSA depends on non-existence of
  such a method.
• Online financial transactions will be compromised
  if such a method is found!

49
Ulams Spirals

• If you draw numbers in a spiral fashion, primes
  tend to concentrate along diagonals.
• More formally, certain quadratic equations (of
  the form am2 bm c) have a very large density
  of primes while some other do not.
• Why?

50

• THANK YOU!