Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Prime Numbers
Bernd Schröder
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Introduction
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Introduction
1. Prime numbers are the atoms of number theory in the sense that
they cannot be decomposed into products of smaller numbers.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Introduction
1. Prime numbers are the atoms of number theory in the sense that
they cannot be decomposed into products of smaller numbers.
(That is part of the definition.)
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Introduction
1. Prime numbers are the atoms of number theory in the sense that
they cannot be decomposed into products of smaller numbers.
(That is part of the definition.)
2. And in the sense that all numbers are products of prime numbers.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Introduction
1. Prime numbers are the atoms of number theory in the sense that
they cannot be decomposed into products of smaller numbers.
(That is part of the definition.)
2. And in the sense that all numbers are products of prime numbers.
(That’s the Fundamental Theorem of Arithmetic, which will be
discussed in the next presentation.)
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Introduction
1. Prime numbers are the atoms of number theory in the sense that
they cannot be decomposed into products of smaller numbers.
(That is part of the definition.)
2. And in the sense that all numbers are products of prime numbers.
(That’s the Fundamental Theorem of Arithmetic, which will be
discussed in the next presentation.)
3. In this presentation we’ll talk about what prime numbers are and
give some early indications how to find prime numbers.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Introduction
1. Prime numbers are the atoms of number theory in the sense that
they cannot be decomposed into products of smaller numbers.
(That is part of the definition.)
2. And in the sense that all numbers are products of prime numbers.
(That’s the Fundamental Theorem of Arithmetic, which will be
discussed in the next presentation.)
3. In this presentation we’ll talk about what prime numbers are and
give some early indications how to find prime numbers.
4. We will conclude with some results that we won’t prove, but
which indicate how much (as well as how little) we know about
prime numbers and their distribution within the natural numbers.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Introduction
1. Prime numbers are the atoms of number theory in the sense that
they cannot be decomposed into products of smaller numbers.
(That is part of the definition.)
2. And in the sense that all numbers are products of prime numbers.
(That’s the Fundamental Theorem of Arithmetic, which will be
discussed in the next presentation.)
3. In this presentation we’ll talk about what prime numbers are and
give some early indications how to find prime numbers.
4. We will conclude with some results that we won’t prove, but
which indicate how much (as well as how little) we know about
prime numbers and their distribution within the natural numbers.
5. From here on, the facts that it is hard to check if large numbers
are prime or not, and that it is hard to find large prime numbers
will drive most of this course.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Definition (reminder).
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Definition (reminder). A prime number is an integer greater than 1
that is only divisible by 1 and by itself.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Definition (reminder). A prime number is an integer greater than 1
that is only divisible by 1 and by itself.
Integers greater than 1 that are not prime are called composite.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Lemma.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Lemma. For every integer greater than 1, there is a prime number
that divides it.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Lemma. For every integer greater than 1, there is a prime number
that divides it.
Proof.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Lemma. For every integer greater than 1, there is a prime number
that divides it.
Proof. Strong induction on n.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Lemma. For every integer greater than 1, there is a prime number
that divides it.
Proof. Strong induction on n.
Base step, n = 1.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Lemma. For every integer greater than 1, there is a prime number
that divides it.
Proof. Strong induction on n.
Base step, n = 1. Trivial.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Lemma. For every integer greater than 1, there is a prime number
that divides it.
Proof. Strong induction on n.
Base step, n = 1. Trivial.
Induction step, {1, . . . , n − 1} → n.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Lemma. For every integer greater than 1, there is a prime number
that divides it.
Proof. Strong induction on n.
Base step, n = 1. Trivial.
Induction step, {1, . . . , n − 1} → n. Let n > 1 be an integer.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Lemma. For every integer greater than 1, there is a prime number
that divides it.
Proof. Strong induction on n.
Base step, n = 1. Trivial.
Induction step, {1, . . . , n − 1} → n. Let n > 1 be an integer. If n is
prime, then n|n.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Lemma. For every integer greater than 1, there is a prime number
that divides it.
Proof. Strong induction on n.
Base step, n = 1. Trivial.
Induction step, {1, . . . , n − 1} → n. Let n > 1 be an integer. If n is
prime, then n|n. If n is composite, then n = pq, where 1 < p < n
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Lemma. For every integer greater than 1, there is a prime number
that divides it.
Proof. Strong induction on n.
Base step, n = 1. Trivial.
Induction step, {1, . . . , n − 1} → n. Let n > 1 be an integer. If n is
prime, then n|n. If n is composite, then n = pq, where 1 < p < n and,
by induction hypothesis, there is a prime number that divides p.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Lemma. For every integer greater than 1, there is a prime number
that divides it.
Proof. Strong induction on n.
Base step, n = 1. Trivial.
Induction step, {1, . . . , n − 1} → n. Let n > 1 be an integer. If n is
prime, then n|n. If n is composite, then n = pq, where 1 < p < n and,
by induction hypothesis, there is a prime number that divides p.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. There are infinitely many prime numbers.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. There are infinitely many prime numbers.
Proof.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. There are infinitely many prime numbers.
Proof. Suppose for a contradiction that there are finitely many, say, n,
prime numbers.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. There are infinitely many prime numbers.
Proof. Suppose for a contradiction that there are finitely many, say, n,
prime numbers. Call them p1 , . . . , pn .
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. There are infinitely many prime numbers.
Proof. Suppose for a contradiction that there are finitely many, say, n,
prime numbers. Call them p1 , . . . , pn . Then one of them, say, pj ,
n
divides ∏ pi + 1.
i=1
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. There are infinitely many prime numbers.
Proof. Suppose for a contradiction that there are finitely many, say, n,
prime numbers. Call them p1 , . . . , pn . Then one of them, say, pj ,
n
n
divides ∏ pi + 1. But then, because pj divides ∏ pi
i=1
Bernd Schröder
Prime Numbers
i=1
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. There are infinitely many prime numbers.
Proof. Suppose for a contradiction that there are finitely many, say, n,
prime numbers. Call them p1 , . . . , pn . Then one of them, say, pj ,
n
n
divides ∏ pi + 1. But then, because pj divides ∏ pi , we conclude that
i=1
i=1
pj |1, which cannot be.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. There are infinitely many prime numbers.
Proof. Suppose for a contradiction that there are finitely many, say, n,
prime numbers. Call them p1 , . . . , pn . Then one of them, say, pj ,
n
n
divides ∏ pi + 1. But then, because pj divides ∏ pi , we conclude that
i=1
i=1
pj |1, which cannot be.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. If n is a composite integer, then there is a prime number
√
≤ n that divides it.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. If n is a composite integer, then there is a prime number
√
≤ n that divides it.
Proof.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. If n is a composite integer, then there is a prime number
√
≤ n that divides it.
Proof. Suppose for a contradiction, that all prime numbers that divide
√
n are greater than n.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. If n is a composite integer, then there is a prime number
√
≤ n that divides it.
Proof. Suppose for a contradiction, that all prime numbers that divide
√
n are greater than n. Because n is composite, n = ab for some
integers 1 < a, b < n.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. If n is a composite integer, then there is a prime number
√
≤ n that divides it.
Proof. Suppose for a contradiction, that all prime numbers that divide
√
n are greater than n. Because n is composite, n = ab for some
integers 1 < a, b < n. Let pa be a prime number that divides a
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. If n is a composite integer, then there is a prime number
√
≤ n that divides it.
Proof. Suppose for a contradiction, that all prime numbers that divide
√
n are greater than n. Because n is composite, n = ab for some
integers 1 < a, b < n. Let pa be a prime number that divides a and let
pb be a prime number that divides b.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. If n is a composite integer, then there is a prime number
√
≤ n that divides it.
Proof. Suppose for a contradiction, that all prime numbers that divide
√
n are greater than n. Because n is composite, n = ab for some
integers 1 < a, b < n. Let pa be a prime number that divides a and let
pb be a prime number that divides b. Then pa and pb are prime factors
√
of n, so pa , pb > n.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. If n is a composite integer, then there is a prime number
√
≤ n that divides it.
Proof. Suppose for a contradiction, that all prime numbers that divide
√
n are greater than n. Because n is composite, n = ab for some
integers 1 < a, b < n. Let pa be a prime number that divides a and let
pb be a prime number that divides b. Then pa and pb are prime factors
√
of n, so pa , pb > n. But then n ≥ pa pb
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. If n is a composite integer, then there is a prime number
√
≤ n that divides it.
Proof. Suppose for a contradiction, that all prime numbers that divide
√
n are greater than n. Because n is composite, n = ab for some
integers 1 < a, b < n. Let pa be a prime number that divides a and let
pb be a prime number that divides b. Then pa and pb are prime factors
√ √
√
of n, so pa , pb > n. But then n ≥ pa pb > n n
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. If n is a composite integer, then there is a prime number
√
≤ n that divides it.
Proof. Suppose for a contradiction, that all prime numbers that divide
√
n are greater than n. Because n is composite, n = ab for some
integers 1 < a, b < n. Let pa be a prime number that divides a and let
pb be a prime number that divides b. Then pa and pb are prime factors
√ √
√
of n, so pa , pb > n. But then n ≥ pa pb > n n = n
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. If n is a composite integer, then there is a prime number
√
≤ n that divides it.
Proof. Suppose for a contradiction, that all prime numbers that divide
√
n are greater than n. Because n is composite, n = ab for some
integers 1 < a, b < n. Let pa be a prime number that divides a and let
pb be a prime number that divides b. Then pa and pb are prime factors
√ √
√
of n, so pa , pb > n. But then n ≥ pa pb > n n = n, which is a
contradiction.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. If n is a composite integer, then there is a prime number
√
≤ n that divides it.
Proof. Suppose for a contradiction, that all prime numbers that divide
√
n are greater than n. Because n is composite, n = ab for some
integers 1 < a, b < n. Let pa be a prime number that divides a and let
pb be a prime number that divides b. Then pa and pb are prime factors
√ √
√
of n, so pa , pb > n. But then n ≥ pa pb > n n = n, which is a
contradiction.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
1. Go through all numbers consecutively.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
1. Go through all numbers consecutively.
2. Check each number if it is divisible by any of the prime numbers
discovered so far.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
1. Go through all numbers consecutively.
2. Check each number if it is divisible by any of the prime numbers
discovered so far.
3. If so, then the number is not prime, and we go to the next
number.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
1. Go through all numbers consecutively.
2. Check each number if it is divisible by any of the prime numbers
discovered so far.
3. If so, then the number is not prime, and we go to the next
number.
4. If not, then the number is prime.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
1. Go through all numbers consecutively.
2. Check each number if it is divisible by any of the prime numbers
discovered so far.
3. If so, then the number is not prime, and we go to the next
number.
4. If not, then the number is prime. We record it as a prime number
and go to the next number.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
1. Go through all numbers consecutively.
2. Check each number if it is divisible by any of the prime numbers
discovered so far.
3. If so, then the number is not prime, and we go to the next
number.
4. If not, then the number is prime. We record it as a prime number
and go to the next number.
5. Even numbers greater than 2 need not be checked.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
1. Go through all numbers consecutively.
2. Check each number if it is divisible by any of the prime numbers
discovered so far.
3. If so, then the number is not prime, and we go to the next
number.
4. If not, then the number is prime. We record it as a prime number
and go to the next number.
5. Even numbers greater than 2 need not be checked.
6. When checking k, primes p < k with p2 > k need not be used.
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7, 9
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−
9
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−,
9 11
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−,
9 11, 13
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−,
9 11, 13, 15
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−,
9 11, 13,−−
15
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−,
9 11, 13,−−,
15 17
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−,
9 11, 13,−−,
15 17, 19
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−,
9 11, 13,−−,
15 17, 19, 21
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−,
9 11, 13,−−,
15 17, 19,−−
21
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−,
9 11, 13,−−,
15 17, 19,−−,
21 23
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−,
9 11, 13,−−,
15 17, 19,−−,
21 23, 25
Bernd Schröder
Prime Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−,
9 11, 13,−−,
15 17, 19,−−,
21 23,−−
25
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Prime Numbers
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Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−,
9 11, 13,−−,
15 17, 19,−−,
21 23,−−,
25 27
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−,
9 11, 13,−−,
15 17, 19,−−,
21 23,−−,
25 −−
27
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−,
9 11, 13,−−,
15 17, 19,−−,
21 23,−−,
25 −−,
27 29
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−,
9 11, 13,−−,
15 17, 19,−−,
21 23,−−,
25 −−,
27 29, 31
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−,
9 11, 13,−−,
15 17, 19,−−,
21 23,−−,
25 −−,
27 29, 31, 33
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−,
9 11, 13,−−,
15 17, 19,−−,
21 23,−−,
25 −−,
27 29, 31,−−
33
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−,
9 11, 13,−−,
15 17, 19,−−,
21 23,−−,
25 −−,
27 29, 31,−−,
33 . . .
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−,
9 11, 13,−−,
15 17, 19,−−,
21 23,−−,
25 −−,
27 29, 31,−−,
33 . . .
The larger the number, the longer the algorithm takes.
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−,
9 11, 13,−−,
15 17, 19,−−,
21 23,−−,
25 −−,
27 29, 31,−−,
33 . . .
The larger the number, the longer the algorithm takes. So to check if
8675309 is prime would take a long time in this fashion.
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−,
9 11, 13,−−,
15 17, 19,−−,
21 23,−−,
25 −−,
27 29, 31,−−,
33 . . .
The larger the number, the longer the algorithm takes. So to check if
8675309 is prime would take a long time in this fashion. (BTW, it is.)
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Sieve of Eratosthenes
2, 3, 5, 7,−−,
9 11, 13,−−,
15 17, 19,−−,
21 23,−−,
25 −−,
27 29, 31,−−,
33 . . .
The larger the number, the longer the algorithm takes. So to check if
8675309 is prime would take a long time in this fashion. (BTW, it is.)
One of the problems is that we need to create the prime numbers
before we can use them for the test.
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Finding Prime Numbers
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Finding Prime Numbers
1. The sieve of Eratosthenes takes O n2 operations to determine if
n is prime.
Bernd Schröder
Prime Numbers
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Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Finding Prime Numbers
1. The sieve of Eratosthenes takes O n2 operations to determine if
n is prime. For numbers with dozens of digits, this is much too
long.
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Finding Prime Numbers
1. The sieve of Eratosthenes takes O n2 operations to determine if
n is prime. For numbers with dozens of digits, this is much too
long.
2. There is no useful formula to generate prime numbers.
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Finding Prime Numbers
1. The sieve of Eratosthenes takes O n2 operations to determine if
n is prime. For numbers with dozens of digits, this is much too
long.
2. There is no useful formula to generate prime numbers. (Such a
formula was a dream of Gauss’, and even someone as brilliant as
him did not succeed.)
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Finding Prime Numbers
1. The sieve of Eratosthenes takes O n2 operations to determine if
n is prime. For numbers with dozens of digits, this is much too
long.
2. There is no useful formula to generate prime numbers. (Such a
formula was a dream of Gauss’, and even someone as brilliant as
him did not succeed.)
3. There is an algorithm that
determines if a given number n is
prime in O (log(n))12 bit operations.
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Finding Prime Numbers
1. The sieve of Eratosthenes takes O n2 operations to determine if
n is prime. For numbers with dozens of digits, this is much too
long.
2. There is no useful formula to generate prime numbers. (Such a
formula was a dream of Gauss’, and even someone as brilliant as
him did not succeed.)
3. There is an algorithm that
determines if a given number n is
prime in O (log(n))12 bit operations. The exponent can be
reduced to 6 + ε.
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Finding Prime Numbers
1. The sieve of Eratosthenes takes O n2 operations to determine if
n is prime. For numbers with dozens of digits, this is much too
long.
2. There is no useful formula to generate prime numbers. (Such a
formula was a dream of Gauss’, and even someone as brilliant as
him did not succeed.)
3. There is an algorithm that
determines if a given number n is
prime in O (log(n))12 bit operations. The exponent can be
reduced to 6 + ε. (We will encounter a probabilistic algorithm,
which has similar efficiency, but which has not been proved to
always give the right answer.)
Bernd Schröder
Prime Numbers
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Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Function That Counts Prime Numbers
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Function That Counts Prime Numbers
Definition.
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Prime Numbers
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Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Function That Counts Prime Numbers
Definition. For every x ∈ (1, ∞) let π(x) be the number of prime
numbers that are less than or equal to x.
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Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Function That Counts Prime Numbers
Definition. For every x ∈ (1, ∞) let π(x) be the number of prime
numbers that are less than or equal to x.
If we know where the jumps of π(x) are, we know where the prime
numbers are.
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Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Function That Counts Prime Numbers
Definition. For every x ∈ (1, ∞) let π(x) be the number of prime
numbers that are less than or equal to x.
If we know where the jumps of π(x) are, we know where the prime
numbers are. So (pipe dream) if we had an easy-to-compute function
that is within 12 of π(x), we would know where all the prime numbers
are.
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Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Function That Counts Prime Numbers
Definition. For every x ∈ (1, ∞) let π(x) be the number of prime
numbers that are less than or equal to x.
If we know where the jumps of π(x) are, we know where the prime
numbers are. So (pipe dream) if we had an easy-to-compute function
that is within 12 of π(x), we would know where all the prime numbers
are.
Theorem.
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Prime Numbers
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Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
The Function That Counts Prime Numbers
Definition. For every x ∈ (1, ∞) let π(x) be the number of prime
numbers that are less than or equal to x.
If we know where the jumps of π(x) are, we know where the prime
numbers are. So (pipe dream) if we had an easy-to-compute function
that is within 12 of π(x), we would know where all the prime numbers
are.
Theorem. Prime Number
Z x Theorem
1
x
π(x) = O (li(x)) = O
du = O
ln(x)
2 ln(u)
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Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Corollary.
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Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Corollary. If pn is the nth prime number, then pn = O(n log(n)).
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Prime Numbers
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Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Corollary. If pn is the nth prime number, then pn = O(n log(n)).
Proof.
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Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Corollary. If pn is the nth prime number, then pn = O(n log(n)).
Proof. By the prime number theorem, the number of prime numbers
in the interval interval
[1, n log(n)) is
n log(n)
O
ln(n log(n))
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Prime Numbers
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Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Corollary. If pn is the nth prime number, then pn = O(n log(n)).
Proof. By the prime number theorem, the number of prime numbers
in the interval interval
[1,
n log(n)) is
n log(n)
n log(n)
O
=O
ln(n log(n))
ln(n) + ln(log(n))
Bernd Schröder
Prime Numbers
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Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Corollary. If pn is the nth prime number, then pn = O(n log(n)).
Proof. By the prime number theorem, the number of prime numbers
in the interval interval
[1,
n log(n)) is
n log(n)
n log(n)
O
=O
= O (n) .
ln(n log(n))
ln(n) + ln(log(n))
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Corollary. If pn is the nth prime number, then pn = O(n log(n)).
Proof. By the prime number theorem, the number of prime numbers
in the interval interval
[1,
n log(n)) is
n log(n)
n log(n)
O
=O
= O (n) .
ln(n log(n))
ln(n) + ln(log(n))
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem.
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. For any positive integer n, there are n consecutive
composite integers.
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Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. For any positive integer n, there are n consecutive
composite integers.
Proof.
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Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. For any positive integer n, there are n consecutive
composite integers.
Proof. Let n ∈ N and consider
(n + 1)! + 2, (n + 1)! + 3, . . . , (n + 1)! + (n + 1).
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Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. For any positive integer n, there are n consecutive
composite integers.
Proof. Let n ∈ N and consider
(n + 1)! + 2, (n + 1)! + 3, . . . , (n + 1)! + (n + 1). Then every one of
these numbers is a composite number, because j|(n + 1)! + j for
j = 2, . . . , n + 1.
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Prime Numbers
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Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Theorem. For any positive integer n, there are n consecutive
composite integers.
Proof. Let n ∈ N and consider
(n + 1)! + 2, (n + 1)! + 3, . . . , (n + 1)! + (n + 1). Then every one of
these numbers is a composite number, because j|(n + 1)! + j for
j = 2, . . . , n + 1.
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Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Conjectures About Prime Numbers
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Conjectures About Prime Numbers
1. Bertrand’s Conjecture/Postulate.
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Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Conjectures About Prime Numbers
1. Bertrand’s Conjecture/Postulate. For every positive integer n
there is a prime number p so that n < p < 2n.
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Conjectures About Prime Numbers
1. Bertrand’s Conjecture/Postulate. For every positive integer n
there is a prime number p so that n < p < 2n.
2. Twin Prime Conjecture.
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Conjectures About Prime Numbers
1. Bertrand’s Conjecture/Postulate. For every positive integer n
there is a prime number p so that n < p < 2n.
2. Twin Prime Conjecture. There are infinitely many prime
numbers p so that p + 2 is prime, too.
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Conjectures About Prime Numbers
1. Bertrand’s Conjecture/Postulate. For every positive integer n
there is a prime number p so that n < p < 2n.
2. Twin Prime Conjecture. There are infinitely many prime
numbers p so that p + 2 is prime, too.
3. Erdös Conjecture on Arithmetic Progressions of Prime
Numbers, now the Green-Tao Theorem.
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Conjectures About Prime Numbers
1. Bertrand’s Conjecture/Postulate. For every positive integer n
there is a prime number p so that n < p < 2n.
2. Twin Prime Conjecture. There are infinitely many prime
numbers p so that p + 2 is prime, too.
3. Erdös Conjecture on Arithmetic Progressions of Prime
Numbers, now the Green-Tao Theorem. For every positive
integer n ≥ 3 there is an arithmetic progression
p, p + k, . . . , p + (n − 1)k of n prime numbers.
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Conjectures About Prime Numbers
1. Bertrand’s Conjecture/Postulate. For every positive integer n
there is a prime number p so that n < p < 2n.
2. Twin Prime Conjecture. There are infinitely many prime
numbers p so that p + 2 is prime, too.
3. Erdös Conjecture on Arithmetic Progressions of Prime
Numbers, now the Green-Tao Theorem. For every positive
integer n ≥ 3 there is an arithmetic progression
p, p + k, . . . , p + (n − 1)k of n prime numbers. (Problem with
regards to twin primes
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Conjectures About Prime Numbers
1. Bertrand’s Conjecture/Postulate. For every positive integer n
there is a prime number p so that n < p < 2n.
2. Twin Prime Conjecture. There are infinitely many prime
numbers p so that p + 2 is prime, too.
3. Erdös Conjecture on Arithmetic Progressions of Prime
Numbers, now the Green-Tao Theorem. For every positive
integer n ≥ 3 there is an arithmetic progression
p, p + k, . . . , p + (n − 1)k of n prime numbers. (Problem with
regards to twin primes: k need not be 2.)
Bernd Schröder
Prime Numbers
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Louisiana Tech University, College of Engineering and Science
Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Conjectures About Prime Numbers
1. Bertrand’s Conjecture/Postulate. For every positive integer n
there is a prime number p so that n < p < 2n.
2. Twin Prime Conjecture. There are infinitely many prime
numbers p so that p + 2 is prime, too.
3. Erdös Conjecture on Arithmetic Progressions of Prime
Numbers, now the Green-Tao Theorem. For every positive
integer n ≥ 3 there is an arithmetic progression
p, p + k, . . . , p + (n − 1)k of n prime numbers. (Problem with
regards to twin primes: k need not be 2.)
4. Goldbach’s Conjecture.
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Prime Numbers
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Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Conjectures About Prime Numbers
1. Bertrand’s Conjecture/Postulate. For every positive integer n
there is a prime number p so that n < p < 2n.
2. Twin Prime Conjecture. There are infinitely many prime
numbers p so that p + 2 is prime, too.
3. Erdös Conjecture on Arithmetic Progressions of Prime
Numbers, now the Green-Tao Theorem. For every positive
integer n ≥ 3 there is an arithmetic progression
p, p + k, . . . , p + (n − 1)k of n prime numbers. (Problem with
regards to twin primes: k need not be 2.)
4. Goldbach’s Conjecture. Every even positive integer greater
than 2 is the sum of two prime numbers.
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Prime Numbers
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Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Conjectures About Prime Numbers
1. Bertrand’s Conjecture/Postulate. For every positive integer n
there is a prime number p so that n < p < 2n.
2. Twin Prime Conjecture. There are infinitely many prime
numbers p so that p + 2 is prime, too.
3. Erdös Conjecture on Arithmetic Progressions of Prime
Numbers, now the Green-Tao Theorem. For every positive
integer n ≥ 3 there is an arithmetic progression
p, p + k, . . . , p + (n − 1)k of n prime numbers. (Problem with
regards to twin primes: k need not be 2.)
4. Goldbach’s Conjecture. Every even positive integer greater
than 2 is the sum of two prime numbers.
5. n2 + 1 Conjecture.
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Prime Numbers
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Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Conjectures About Prime Numbers
1. Bertrand’s Conjecture/Postulate. For every positive integer n
there is a prime number p so that n < p < 2n.
2. Twin Prime Conjecture. There are infinitely many prime
numbers p so that p + 2 is prime, too.
3. Erdös Conjecture on Arithmetic Progressions of Prime
Numbers, now the Green-Tao Theorem. For every positive
integer n ≥ 3 there is an arithmetic progression
p, p + k, . . . , p + (n − 1)k of n prime numbers. (Problem with
regards to twin primes: k need not be 2.)
4. Goldbach’s Conjecture. Every even positive integer greater
than 2 is the sum of two prime numbers.
5. n2 + 1 Conjecture. There are infinitely many prime numbers of
the form n2 + 1.
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Prime Numbers
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Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Conjectures About Prime Numbers
1. Bertrand’s Conjecture/Postulate. For every positive integer n
there is a prime number p so that n < p < 2n.
2. Twin Prime Conjecture. There are infinitely many prime
numbers p so that p + 2 is prime, too.
3. Erdös Conjecture on Arithmetic Progressions of Prime
Numbers, now the Green-Tao Theorem. For every positive
integer n ≥ 3 there is an arithmetic progression
p, p + k, . . . , p + (n − 1)k of n prime numbers. (Problem with
regards to twin primes: k need not be 2.)
4. Goldbach’s Conjecture. Every even positive integer greater
than 2 is the sum of two prime numbers.
5. n2 + 1 Conjecture. There are infinitely many prime numbers of
the form n2 + 1.
6. Legendre Conjecture.
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Prime Numbers
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Prime Numbers
The Sieve of Eratosthenes
Large Prime Numbers
The Distribution of Prime Numbers
Conjectures About Prime Numbers
1. Bertrand’s Conjecture/Postulate. For every positive integer n
there is a prime number p so that n < p < 2n.
2. Twin Prime Conjecture. There are infinitely many prime
numbers p so that p + 2 is prime, too.
3. Erdös Conjecture on Arithmetic Progressions of Prime
Numbers, now the Green-Tao Theorem. For every positive
integer n ≥ 3 there is an arithmetic progression
p, p + k, . . . , p + (n − 1)k of n prime numbers. (Problem with
regards to twin primes: k need not be 2.)
4. Goldbach’s Conjecture. Every even positive integer greater
than 2 is the sum of two prime numbers.
5. n2 + 1 Conjecture. There are infinitely many prime numbers of
the form n2 + 1.
6. Legendre Conjecture. Between any two consecutive squares of
integers, there is a prime number.
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Prime Numbers
Louisiana Tech University, College of Engineering and Science