the mystery of prime numbers -...

THE MYSTERY OF PRIME NUMBERS

Sunil K. CheboluIllinois State University

REU Presentation, Summer 2013

1/35 Sunil Chebolu THE MYSTERY OF PRIME NUMBERS

It will be another million years, at least, before we understand theprimes

Paul Erdos


Talk Outline

I Prime Number Theorem

I Riemann Hypothesis

I Primality testing

I Goldbach Conjecture

I Twin Prime Conjecture


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 primenumbers whose difference is bigger than n?

This will be answered in the last slide.


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

limx→∞

π(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 thedata he obtained he stated (in 1790 without proof) that

π(x) ≈ x

ln x


For a long time no one was able to prove this.

Inspired by some ideas introduced by Riemann, Hadamard andPoussin independently proved this in 1896. This is now called thePrime Number Theorem.

In the first half of the 20th century, some mathematicians(including Hardy) believed that there exists a hierarchy of proofmethods in mathematics depending on what sorts of numbers(integers, reals, complex) a proof requires.

In this hierarchy, the prime number theorem (PNT) was believedto be a “deep” theorem by virtue of requiring complex analysis.

This belief was shaken when a completely elementary proof of thePNT was given by Erdos and Selberg in 1948.

! Elementary does not mean Easy !


Prime Number Theorem

π(x) ≈ x

ln x

limx→∞

π(x)

x/ ln x= 1

Roughly speaking, this means that the probability that a randomlychosen number of magnitude x is a prime is 1/ ln(x)

Yet another way to think about the PNT: Let pn denote the n-thprime number, then pn ≈ n ln n


The Riemann Hypothesis

This is widely acknowledged to be the most important unsolvedproblem in mathematics.

The Riemann zeta function is a function of a complex variable zdefined by the series

ζ(z) =∞∑n=1

1

nz

This was studied by Euler when z is real. Euler proved:

ζ(2) =π2

6


Riemann introduced this function in his famous 6-page paper (andhis 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 ideasto the study of prime numbers.

Hadamard and Poussin drew heavily on these ideas in their proofof the PNT.

In this paper Riemann gave a precise formula for π(x) in terms theintegral of 1/log(x) and the zeros of the zeta function!


Figure : Bernhard Riemann

Riemann Hypothesis Riemann (1859) conjectured that all thenon-trivial zeros of the zeta function lie on the line Re(z) = 1/2.


why is this conjecture so important and so interesting?

The Riemann zeta function is the key to understanding the primes.

ζ(z) =∞∑n=1

1

nz=

∏p prime

1

1− p−z

Knowledge about the location of the zeros of this function givesuseful information about the distribution of primes.

For instance, the prime number theorem is equivalent to the factthat 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 isthe distribution of the primes.


A statistical analogy

If the prime number theorem tells us something about the averagedistribution of the primes along the number line, then the Riemannhypothesis tells us something about the deviation from the average.


Riemann computed the first few zeros of the zeta function by handand found that they satisfy his hypothesis. That is, they lie on thecritical line.

Using modern computers over 1.5 billion zeros of the zeta functionhave been computed and they all fell on the critical line. That is avery strong experimental evidence, but not a proof.

In mathematics, we need a proof! Some logical argument whichcan explain why something is true.


The Riemann hypothesis has several remarkable consequences.

There are many statements which are all equivalent to theRiemann hypothesis.

Therefore the person who proves the Riemann hypothesis willprove a hundred theorems at once.

In the year 2000, the Clay Mathematics Institute stated sevenmajor problems in mathematics which are called the MillenniumPrize problems. A correct solution to any of these problems resultsin a cash prize of $ 1,000,000

The Riemann hypothesis is the most famous of these sevenproblems.


Hilbert: If I were to awaken af-ter having slept for a thousandyears, my first question wouldbe:

Has the Riemann hypothesisbeen proven?


Primality Testing

Problem: Given a positive integer n, how can we determineefficiently whether n is a prime number or not?

More precisely, is there a polynomial time algorithm for primailitytesting? (That is, an algorithm in which the number ofarithemetical operations/steps is no more than cdA where c and Aare fixed constants and d is the number of digits of n).


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 primenumbers.

Pure reason: .. the dignity of the science itself seems to requirethat every possible means be explored for the solution of a problemso elegant and so celebrated..

– article 329 of Disquisitiones Airthmetica (1801) by C. F. Gauss


This problem has a very long history. (This is going to be the maintopic for my MAT 410 (Topics in Number Theory) in Fall 2013)

I Sieve of Eratosthenes (300 BC)

I Wilson’s test

I Fermat’s test

I Strassen’s test

I Miller-Rabin test

I Pollard’s test

I 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-Rabintest runs in polynomial time.


Big news in 2002!

Gauss’ dream was finally realized in August 2002 when 3 Indiancomputer scientists (Agrawal, Kayal, Saxena) came up with anunconditional, deterministic, polynomial-time algorithm forprimality testing.

Most shocking was the simplicity and originality of their test..while the “experts had made complicated modifications on existingtests to gain improvements (often involving great ingenuity), theseauthors rethought the direction in which to push the usual ideaswith stunning success.


Figure : Kayal, Saxena and Agrawal

Kayal and Saxena were undergraduate students who did some ofthis research in their undergraduate thesis.

Such a successful outcome from undergraduate students is veryimpressive and should inspire all undergraduates.


The Goldbach Conjecture

In a letter dated June 7, 1742, Prussian mathematician ChristianGoldbach proposed the following amazing conjecture.

I Binary Goldbach Conjecture (BGC) Every even numbergreater than 2 is the sum of two primes.Example: 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7 etc.

I Ternary Goldbach Conjecture (TGC) Every odd numbergreater than 7 is the sum of 3 primes.Example: 9 = 3 + 3 + 3, 11 = 3 + 3 = 5, 13 = 3 + 5 + 5 etc

BCG =⇒ TGC


Figure : Goldbach’s letter to Euler


What is known about the BGC?

Richstein (2001) numerically verified TGC for all even integers upto 4 · 1014.

More recently, it has been verified for all even integers up to 1018

Chen (1973) showed that every sufficiently large even number Ncan be be expressed as

N = p1 + p2p3

where p1, p2 and p3 are all primes.


A lot more is known about the TGC

I Saouter (1995) verified numerically TGC for all odd numbersup to 1020

I Chen and Wang (1989) showed that TGC is true for all oddnumbers 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 assumesthe Generalized Riemann Hypothesis.


More recent results

I Olivier Ramare (1995) showed that every odd number ≥ 5 isthe sum of at most 7 primes.

I Leszek Kaniecki showed every odd integer is a sum of at mostfive primes, under the Riemann Hypothesis.

I Terrance Tao (2012) proved this without the Reimannhypothesis.

I Harald Helfgott (May 2013) proved the Ternary GoldbachConjecture!


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.

I 2007− 20036636132195000 ± 1 (58711 digits)

I 2009− 655164683552333333 ± 1 (100355 digits)

I 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.


A well known result of Leibnitz:

The harmonic series∞∑n=1

1

n

diverges.

A less well-known result (Apostol) The series∑p prime

1

p

diverges.

Note that this result implies that there exists infinitely manyprimes.


What about the series ∑p twin prime

1

p?

Note that if this serives diverges, then the Twin prime conjecture issettled.

However, Brun showed that this coverges!∑p twin prime

1

p

is a convergent series.


Chen showed the following result which sounds pretty close to theTwin prime conjecture

There are infinitely many pairs of integers (p, p + 2) where p is aprime, and p + 2 is a product of at most two primes.

His proof uses complicated techniques from analytic number theory.


A recent breakthrough

Yitan Zhang, a lecturer at the University of New Hampshireannounced the following result in April 2013.

There exists an integer N(< 70 million), for which there areinfinitely 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 ofMathematics.


Zhang’s career (from wikipedia)

Zhang’s Ph.D. work was on the Jacobian conjecture. He originallythought that he had solved the problem but it turned out that hehad not. After graduation, Zhang had a hard time finding anacademic position. In a recent article, Zhang’s thesis advisor,Professor Tzuong-Tsieng Moh, recalled that ”Sometimes Iregretted not fixing him a job” and ”He never came back to merequesting recommendation letters.” He managed to find aposition as a lecturer after many years. He is still currently alecturer at the University of New Hampshire; he worked for severalyears as an accountant, a delivery worker for a New York Cityrestaurant, in a motel in Kentucky and in a Subway sandwich shopbefore working as a lecturer.


Moral of the story

Zhang’s success is very in-spiring. It shows that ma-jor breakthroughs don’t haveto come from topnotch math-ematicians at Harvard andPrinceton. Hard work and per-severance can bring success foranyone.


Answer to my first question

Given a positive integer n does there exists two consecutive primenumbers 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.


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 meansπ(x) ≥ x/n.

why? because there are x/n such intervals less than x . Divide bothsides by x/ ln x to get:

π(x)

x/ ln x≥ x/n

x/ ln x

Now take limits as x goes to infinity of both sides and apply PNT.We get a contradiction

1 ≥ ∞34/35 Sunil Chebolu THE MYSTERY OF PRIME NUMBERS

Thank you!


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