THE MYSTERY OF PRIME NUMBERS
Sunil K. Chebolu
Illinois State University
REU Presentation, Summer 2013
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Sunil Chebolu
THE MYSTERY OF PRIME NUMBERS
It will be another million years, at least, before we understand the
primes
Paul Erdös
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Sunil Chebolu
THE MYSTERY OF PRIME NUMBERS
Talk Outline
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Prime Number Theorem
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Riemann Hypothesis
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Primality testing
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Goldbach Conjecture
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Twin Prime Conjecture
Sunil Chebolu
THE MYSTERY OF PRIME NUMBERS
A fun question
Are there arbitrarily long gaps between consecutive prime numbers?
Here is the list of primes less than 100:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67,
71, 73, 79, 83, 89, 97
The largest gap is 8.
But if we look further we can find larger gaps.
Given a positive integer n does there exist two consecutive prime
numbers whose difference is bigger than n?
This will be answered in the last slide.
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Sunil Chebolu
THE MYSTERY OF PRIME NUMBERS
Prime Counting Function
Let π(x) denote the number of primes less than or equal to x.
For example, π(6) = 3, π(10) = 4, etc.
Here is a well-known result of Euclid
lim π(x) = ∞
x→∞
Is there a nice formula for π(x)?
Some of the very best minds have thought about this question.
Young Gauss in his spare time computed primes and based on the
data he obtained he stated (in 1790 without proof) that
π(x) ≈
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Sunil Chebolu
x
ln x
THE MYSTERY OF PRIME NUMBERS
For a long time no one was able to prove this.
Inspired by some ideas introduced by Riemann, Hadamard and
Poussin independently proved this in 1896. This is now called the
Prime Number Theorem.
In the first half of the 20th century, some mathematicians
(including Hardy) believed that there exists a hierarchy of proof
methods in mathematics depending on what sorts of numbers
(integers, reals, complex) a proof requires.
In this hierarchy, the prime number theorem (PNT) was believed
to be a “deep” theorem by virtue of requiring complex analysis.
This belief was shaken when a completely elementary proof of the
PNT was given by Erdos and Selberg in 1948.
! Elementary does not mean Easy !
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Sunil Chebolu
THE MYSTERY OF PRIME NUMBERS
Prime Number Theorem
π(x) ≈
lim
x→∞
x
ln x
π(x)
=1
x/ ln x
Roughly speaking, this means that the probability that a randomly
chosen number of magnitude x is a prime is 1/ ln(x)
Yet another way to think about the PNT: Let pn denote the n-th
prime number, then pn ≈ n ln n
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Sunil Chebolu
THE MYSTERY OF PRIME NUMBERS
The Riemann Hypothesis
This is widely acknowledged to be the most important unsolved
problem in mathematics.
The Riemann zeta function is a function of a complex variable z
defined by the series
∞
X
1
ζ(z) =
nz
n=1
This was studied by Euler when z is real. Euler proved:
ζ(2) =
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π2
6
THE MYSTERY OF PRIME NUMBERS
Riemann introduced this function in his famous 6-page paper (and
his only paper on number theory!):
Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse =
On the number of primes less than a given quantity
This highly influencial paper introduced some radically new ideas
to the study of prime numbers.
Hadamard and Poussin drew heavily on these ideas in their proof
of the PNT.
In this paper Riemann gave a precise formula for π(x) in terms the
integral of 1/log(x) and the zeros of the zeta function!
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THE MYSTERY OF PRIME NUMBERS
Figure : Bernhard Riemann
Riemann Hypothesis Riemann (1859) conjectured that all the
non-trivial zeros of the zeta function lie on the line Re(z) = 1/2.
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THE MYSTERY OF PRIME NUMBERS
why is this conjecture so important and so interesting?
The Riemann zeta function is the key to understanding the primes.
ζ(z) =
∞
X
1
=
nz
n=1
Y
p prime
1
1 − p −z
Knowledge about the location of the zeros of this function gives
useful information about the distribution of primes.
For instance, the prime number theorem is equivalent to the fact
that there are no zeros of the zeta function on the line Re(z) = 1.
The closer the real part of the zeros lies to 1/2, the more regular is
the distribution of the primes.
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A statistical analogy
If the prime number theorem tells us something about the average
distribution of the primes along the number line, then the Riemann
hypothesis tells us something about the deviation from the average.
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THE MYSTERY OF PRIME NUMBERS
Riemann computed the first few zeros of the zeta function by hand
and found that they satisfy his hypothesis. That is, they lie on the
critical line.
Using modern computers over 1.5 billion zeros of the zeta function
have been computed and they all fell on the critical line. That is a
very strong experimental evidence, but not a proof.
In mathematics, we need a proof! Some logical argument which
can explain why something is true.
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THE MYSTERY OF PRIME NUMBERS
The Riemann hypothesis has several remarkable consequences.
There are many statements which are all equivalent to the
Riemann hypothesis.
Therefore the person who proves the Riemann hypothesis will
prove a hundred theorems at once.
In the year 2000, the Clay Mathematics Institute stated seven
major problems in mathematics which are called the Millennium
Prize problems. A correct solution to any of these problems results
in a cash prize of $ 1,000,000
The Riemann hypothesis is the most famous of these seven
problems.
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THE MYSTERY OF PRIME NUMBERS
Hilbert: If I were to awaken after having slept for a thousand
years, my first question would
be:
Has the Riemann hypothesis
been proven?
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THE MYSTERY OF PRIME NUMBERS
Primality Testing
Problem: Given a positive integer n, how can we determine
efficiently whether n is a prime number or not?
More precisely, is there a polynomial time algorithm for primaility
testing? (That is, an algorithm in which the number of
arithemetical operations/steps is no more than cd A where c and A
are fixed constants and d is the number of digits of n).
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THE MYSTERY OF PRIME NUMBERS
Why do we care?
I will give you two reasons:
Applied reason: This is very important in public key cryptosystems
(RSA) which use integers that are product of two large prime
numbers.
Pure reason: .. the dignity of the science itself seems to require
that every possible means be explored for the solution of a problem
so elegant and so celebrated..
– article 329 of Disquisitiones Airthmetica (1801) by C. F. Gauss
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THE MYSTERY OF PRIME NUMBERS
This problem has a very long history. (This is going to be the main
topic for my MAT 410 (Topics in Number Theory) in Fall 2013)
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Sieve of Eratosthenes (300 BC)
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Wilson’s test
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Fermat’s test
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Strassen’s test
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Miller-Rabin test
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Pollard’s test
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Elliptic curve test (1990)
Alas.. none of these tests run in polynomial time.
If you assume the RH, then it can be shown that the Miller-Rabin
test runs in polynomial time.
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THE MYSTERY OF PRIME NUMBERS
Big news in 2002!
Gauss’ dream was finally realized in August 2002 when 3 Indian
computer scientists (Agrawal, Kayal, Saxena) came up with an
unconditional, deterministic, polynomial-time algorithm for
primality testing.
Most shocking was the simplicity and originality of their test..
while the “experts had made complicated modifications on existing
tests to gain improvements (often involving great ingenuity), these
authors rethought the direction in which to push the usual ideas
with stunning success.
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THE MYSTERY OF PRIME NUMBERS
Figure : Kayal, Saxena and Agrawal
Kayal and Saxena were undergraduate students who did some of
this research in their undergraduate thesis.
Such a successful outcome from undergraduate students is very
impressive and should inspire all undergraduates.
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THE MYSTERY OF PRIME NUMBERS
The Goldbach Conjecture
In a letter dated June 7, 1742, Prussian mathematician Christian
Goldbach proposed the following amazing conjecture.
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Binary Goldbach Conjecture (BGC) Every even number
greater than 2 is the sum of two primes.
Example: 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7 etc.
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Ternary Goldbach Conjecture (TGC) Every odd number
greater than 7 is the sum of 3 primes.
Example: 9 = 3 + 3 + 3, 11 = 3 + 3 = 5, 13 = 3 + 5 + 5 etc
BCG =⇒ TGC
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Figure : Goldbach’s letter to Euler
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THE MYSTERY OF PRIME NUMBERS
What is known about the BGC?
Richstein (2001) numerically verified TGC for all even integers up
to 4 · 1014 .
More recently, it has been verified for all even integers up to 1018
Chen (1973) showed that every sufficiently large even number N
can be be expressed as
N = p1 + p2 p3
where p1 , p2 and p3 are all primes.
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THE MYSTERY OF PRIME NUMBERS
A lot more is known about the TGC
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Saouter (1995) verified numerically TGC for all odd numbers
up to 1020
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Chen and Wang (1989) showed that TGC is true for all odd
numbers greater than 1043000 .
What about the odd numbers in between? [1020 , 1043000 ]?
It is shown that the TGC is true for these integers if one assumes
the Generalized Riemann Hypothesis.
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THE MYSTERY OF PRIME NUMBERS
More recent results
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Olivier Ramare (1995) showed that every odd number ≥ 5 is
the sum of at most 7 primes.
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Leszek Kaniecki showed every odd integer is a sum of at most
five primes, under the Riemann Hypothesis.
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Terrance Tao (2012) proved this without the Reimann
hypothesis.
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Harald Helfgott (May 2013) proved the Ternary Goldbach
Conjecture!
Sunil Chebolu
THE MYSTERY OF PRIME NUMBERS
Twin prime conjecture
A pair (p, p + 2) is a twin prime if both p and p + 2 are prime.
Example (3, 5), (5, 7), (11, 13), (17, 19), (1997, 1999) etc.
Mad race in search for large twin primes.
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2007 − 20036636132195000 ± 1 (58711 digits)
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2009 − 655164683552333333 ± 1 (100355 digits)
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2011 − 37568016956852666669 ± 1 (200700)
There are 808,675,888,577,436 twin prime pairs below 1018
It is natural to ask if there exists infinitely many twin primes.
Conjecture: There exists infinitely many twin primes.
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Sunil Chebolu
THE MYSTERY OF PRIME NUMBERS
A well known result of Leibnitz:
The harmonic series
∞
X
1
n=1
n
diverges.
A less well-known result (Apostol) The series
X
p prime
1
p
diverges.
Note that this result implies that there exists infinitely many
primes.
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THE MYSTERY OF PRIME NUMBERS
What about the series
X
p twin prime
1
?
p
Note that if this serives diverges, then the Twin prime conjecture is
settled.
However, Brun showed that this coverges!
X
p twin prime
1
p
is a convergent series.
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THE MYSTERY OF PRIME NUMBERS
Chen showed the following result which sounds pretty close to the
Twin prime conjecture
There are infinitely many pairs of integers (p, p + 2) where p is a
prime, and p + 2 is a product of at most two primes.
His proof uses complicated techniques from analytic number theory.
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THE MYSTERY OF PRIME NUMBERS
A recent breakthrough
Yitan Zhang, a lecturer at the University of New Hampshire
announced the following result in April 2013.
There exists an integer N(< 70 million), for which there are
infinitely many pairs (p, p + N), where both p and p + N are prime.
Yang’s paper was accepted in early May 2013 by the Annals of
Mathematics.
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THE MYSTERY OF PRIME NUMBERS
Zhang’s career (from wikipedia)
Zhang’s Ph.D. work was on the Jacobian conjecture. He originally
thought that he had solved the problem but it turned out that he
had not. After graduation, Zhang had a hard time finding an
academic position. In a recent article, Zhang’s thesis advisor,
Professor Tzuong-Tsieng Moh, recalled that ”Sometimes I
regretted not fixing him a job” and ”He never came back to me
requesting recommendation letters.” He managed to find a
position as a lecturer after many years. He is still currently a
lecturer at the University of New Hampshire; he worked for several
years as an accountant, a delivery worker for a New York City
restaurant, in a motel in Kentucky and in a Subway sandwich shop
before working as a lecturer.
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THE MYSTERY OF PRIME NUMBERS
Moral of the story
Zhang’s success is very inspiring.
It shows that major breakthroughs don’t have
to come from topnotch mathematicians at Harvard and
Princeton. Hard work and perseverance can bring success for
anyone.
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THE MYSTERY OF PRIME NUMBERS
Answer to my first question
Given a positive integer n does there exists two consecutive prime
numbers whose difference is bigger than n?
Consider the following sequece of n consecutive numbers
(n + 1)! + 2, (n + 1)! + 3, (n + 1)! + 4, · · · (n + 1)! + (n + 1)
None of these numbers is a prime. why?
This sequence produces a prime gap > n.
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THE MYSTERY OF PRIME NUMBERS
A solution using the PNT
Suppose the answer to our question is No.
Then there exists a prime in each of the following intervals:
[1, n], [n + 1, 2n], [2n + 1, 3n], [3n + 1, 4n], and so on..
This means
π(x) ≥ x/n.
why? because there are x/n such intervals less than x. Divide both
sides by x/ ln x to get:
π(x)
x/n
≥
x/ ln x
x/ ln x
Now take limits as x goes to infinity of both sides and apply PNT.
We get a contradiction
1≥∞
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Sunil Chebolu
THE MYSTERY OF PRIME NUMBERS
Thank you!
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THE MYSTERY OF PRIME NUMBERS