Prime TimePrime Time

2 3 5 7 11

13 17

19 23

29 31

37

41

A prime number (or prime integer, A prime number (or prime integer, often simply called a "prime" for often simply called a "prime" for short) is a short) is a positive integerpositive integer p>1 that p>1 that has no positive integer has no positive integer divisorsdivisors other other than 1 and itself. than 1 and itself.

The number 1 is a special case which The number 1 is a special case which is considered neither prime nor is considered neither prime nor composite. Although the number 1 composite. Although the number 1 used to be considered a prime, it used to be considered a prime, it requires special treatment in so requires special treatment in so many definitions and applications many definitions and applications involving primes greater than or involving primes greater than or equal to 2 that it is usually placed equal to 2 that it is usually placed into a class of its own. into a class of its own.

With 1 excluded, the smallest prime With 1 excluded, the smallest prime is therefore 2. However, since 2 is is therefore 2. However, since 2 is the only the only even primeeven prime (which, (which, ironically, in some sense makes it the ironically, in some sense makes it the "oddest" prime), it is also somewhat "oddest" prime), it is also somewhat special, and the set of all primes special, and the set of all primes excluding 2 is therefore called the "excluding 2 is therefore called the "odd primesodd primes." ."

The The nnthth prime number is commonly prime number is commonly denoted pdenoted pnn, so p, so p11 = 2, p = 2, p22 = 3, and so = 3, and so on, and may be computed in on, and may be computed in MathematicaMathematica as as PrimePrime[[nn]. ].

The The setset of primes is sometimes of primes is sometimes denoted denoted ..

Euler commented "Mathematicians Euler commented "Mathematicians have tried in vain to this day to have tried in vain to this day to discover some order in the sequence discover some order in the sequence of prime numbers, and we have of prime numbers, and we have reason to believe that it is a mystery reason to believe that it is a mystery into which the mind will never into which the mind will never penetrate." penetrate."

In a 1975 lecture, D. Zagier commented In a 1975 lecture, D. Zagier commented "There are two facts about the distribution "There are two facts about the distribution of prime numbers of which I hope to of prime numbers of which I hope to convince you so overwhelmingly that they convince you so overwhelmingly that they will be permanently engraved in your will be permanently engraved in your hearts. The first is that, despite their simple hearts. The first is that, despite their simple definition and role as the building blocks of definition and role as the building blocks of the natural numbers, the prime numbers the natural numbers, the prime numbers grow like weeds among the natural grow like weeds among the natural numbers, seeming to obey no other law numbers, seeming to obey no other law than that of chance, and nobody can predict than that of chance, and nobody can predict where the next one will sprout. The second where the next one will sprout. The second fact is even more astonishing, for it states fact is even more astonishing, for it states just the opposite: that the prime numbers just the opposite: that the prime numbers exhibit stunning regularity, that there are exhibit stunning regularity, that there are laws governing their behavior, and that laws governing their behavior, and that they obey these laws with almost military they obey these laws with almost military precision" precision"

The The fundamental theorem of arithmeticfundamental theorem of arithmetic states that any states that any positive integerpositive integer (greater than 1) can be represented in (greater than 1) can be represented in exactly oneexactly one way as a way as a productproduct of of primes. primes.

12 = 212 = 222··33 96= 96= 299 = 299 =

The simplest method of finding factors is The simplest method of finding factors is so-called "so-called "direct search factorizationdirect search factorization" " (a.k.a. (a.k.a. trial divisiontrial division). In this method, all ). In this method, all possible factors are systematically possible factors are systematically tested using trial division to see if they tested using trial division to see if they actually actually dividedivide the given number. It is the given number. It is practical only for very small numbers. practical only for very small numbers. More general (and complicated) More general (and complicated) methods include the methods include the elliptic curve factorization methodelliptic curve factorization method and and number field sievenumber field sieve factorization method. factorization method.

Sieve of EratosthenesSieve of Eratosthenes

(who also calculated the (who also calculated the circumference of the (round) earth).circumference of the (round) earth).

Pick the first prime.Pick the first prime.– Cross off every multiple of that prime.Cross off every multiple of that prime.

Pick the next prime (the next number Pick the next prime (the next number not crossed off).not crossed off).– Cross off every multiple thereofCross off every multiple thereof

Continue Continue ad infinitumad infinitum Wolfram DemonstrationWolfram Demonstration

Infinitude of PrimesInfinitude of Primes

Proved by Euclid, Book IX, proposition Proved by Euclid, Book IX, proposition 20: 20: Prime numbers are more than any Prime numbers are more than any assigned multitude of prime numbers.assigned multitude of prime numbers.

Prove by contradiction:Prove by contradiction: Assume P is the largest prime… then Assume P is the largest prime… then

multiply:multiply: 2x3x5x7x11x…P. Add 1.2x3x5x7x11x…P. Add 1. Any divisor will leave a remainder of 1, Any divisor will leave a remainder of 1,

therefore there is no largest prime P.therefore there is no largest prime P.

Euclid’s ProofEuclid’s Proof Theorem.Theorem.

– There are more primes than found in any finite list There are more primes than found in any finite list of primes. of primes.

Proof.Proof. – Call the primes in our finite list Call the primes in our finite list pp11, , pp22, ..., , ..., pprr.  Let .  Let PP

be any common multiple of these primes plus one be any common multiple of these primes plus one (for example, (for example, PP = = pp11pp22......pprr+1).  Now +1).  Now PP is either is either prime or it is not.  If it is prime, then P is a prime prime or it is not.  If it is prime, then P is a prime that was not in our list.  If that was not in our list.  If PP is not prime, then it is is not prime, then it is divisible by some prime, call it divisible by some prime, call it p.  p.  Notice Notice pp can not can not be any of be any of pp11, , pp22, ..., , ..., pprr, otherwise , otherwise pp would divide 1, would divide 1, which is impossible.  So this prime which is impossible.  So this prime pp is some prime is some prime that was not in our original list.  Either way, the that was not in our original list.  Either way, the original list was incomplete. original list was incomplete.

Note that what is found in this proof Note that what is found in this proof is another prime--one not in the is another prime--one not in the given initial set.  There is no size given initial set.  There is no size restriction on this new prime, it may restriction on this new prime, it may even be smaller than some of those even be smaller than some of those in the initial set. in the initial set. 

Prime CuriositiesPrime Curiosities

With the exception of 2 and 3, all With the exception of 2 and 3, all primes are of the form p = 6n primes are of the form p = 6n ± 1± 1, , i.e., i.e., p = 1, 5(mod6)p = 1, 5(mod6)

The function that gives the number The function that gives the number of primes less than or equal to a of primes less than or equal to a number number nn is denoted is denoted (n) and is (n) and is called the called the prime counting functionprime counting function. . The theorem giving an asymptotic The theorem giving an asymptotic form for form for (n) is called the (n) is called the prime number theoremprime number theorem. .

(n) and p(n) and pnn are inverse functions, so are inverse functions, so (p(pnn) = n) = nfor all positive integers and for all positive integers and p p(n)(n) = n = n iffiff n is a prime number. n is a prime number.

Mersenne PrimeMersenne Prime A Mersenne prime is a A Mersenne prime is a MersenneMersenne number number, ,

i.e., a number of the form i.e., a number of the form MMnn = 2 = 2nn – 1 – 1that is that is primeprime. .

In order for MIn order for Mnn to be to be primeprime, , nn must itself be must itself be primeprime. This is true since for . This is true since for compositecomposite n n with factors r and s , n=rs. Therefore, 2with factors r and s , n=rs. Therefore, 2nn - 1 - 1 can be written as 2can be written as 2rsrs - 1 , which is a - 1 , which is a binomial numberbinomial number that always has a factor that always has a factor (2(2rr – 1). – 1).

The first few Mersenne primes are 3, 7, 31, The first few Mersenne primes are 3, 7, 31, 127, 8191, corresponding to indices 2, 3, 5, 127, 8191, corresponding to indices 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, ... 7, 13, 17, 19, 31, 61, 89, ...

Twin PrimesTwin Primes While Hardy and Wright note that "the While Hardy and Wright note that "the

evidence, when examined in detail, appears to evidence, when examined in detail, appears to justify the conjecture," and Shanks states even justify the conjecture," and Shanks states even more strongly, "the evidence is overwhelming," more strongly, "the evidence is overwhelming," Hardy and Wright also note that the proof or Hardy and Wright also note that the proof or disproof of conjectures of this type "is at disproof of conjectures of this type "is at present beyond the resources of mathematics." present beyond the resources of mathematics."

Arenstorf (2004) published a purported proof of Arenstorf (2004) published a purported proof of the conjecture. Unfortunately, a serious error the conjecture. Unfortunately, a serious error was found in the proof. As a result, the paper was found in the proof. As a result, the paper was retracted and the twin prime conjecture was retracted and the twin prime conjecture remains fully open. remains fully open.

Twin Prime ConjectureTwin Prime Conjecture

There are an infinite number of twin primes p, There are an infinite number of twin primes p, p+2p+2

Sophie Germain primes (p & 2p+1) are closely Sophie Germain primes (p & 2p+1) are closely related.related.

The Hardy-Littlewood conjecture states that the The Hardy-Littlewood conjecture states that the number of twin primes number of twin primes x approaches… x approaches…

Where Where 22 is the twin prime constant, is the twin prime constant, ~ .6601618~ .6601618

x

x

dxx

2 222 ))(ln(2~)(

Bertrand’s PostulateBertrand’s Postulate

For For nn > 1, there is always one prime > 1, there is always one prime pp such that such that nn < < pp < < 2n2n..

Proved independently by Proved independently by – Pafnuty ChebychevPafnuty Chebychev– Srinivasa RamanujanSrinivasa Ramanujan– Paul ErdPaul Erdööss

The Prime Number TheoremThe Prime Number Theorem

The prime number theorem gives an The prime number theorem gives an asymptotic form for the prime asymptotic form for the prime counting function counting function (n) , which counts (n) , which counts the number of primes less than some the number of primes less than some integer n . integer n .

)ln(a

a

Gauss (Gauss (GaußGauß))

At age 15 had access to a table (by At age 15 had access to a table (by Lambert) of primes up to a million. Also Lambert) of primes up to a million. Also had a table of logs. Looking at the pair, had a table of logs. Looking at the pair, he said…he said…

Primzahlen unter a(=Primzahlen unter a(=∞∞) a/la) a/la Or, “the number of primes under Or, “the number of primes under aa (as (as aa

approaches infinity) is approximatelyapproaches infinity) is approximately

Legendre’s ImprovementLegendre’s Improvement

Brought Gauss’s approximation up Brought Gauss’s approximation up towards the true number of primes.towards the true number of primes.

08366.1)log()(

N

Nx

Logarithmic IntegralLogarithmic Integral

Gauss later refined his result toGauss later refined his result to

Calculate Li(100) and Li(10,000) Calculate Li(100) and Li(10,000)

x

duu

xLix2 ln

1)()(

For small n, it had been checked and For small n, it had been checked and always found that always found that (n) < Li(n). As a (n) < Li(n). As a result, many prominent mathematicians, result, many prominent mathematicians, including no less than both Gauss and including no less than both Gauss and Riemann, conjectured that the inequality Riemann, conjectured that the inequality was strict. To everyone's surprise, this was strict. To everyone's surprise, this conjecture was refuted when Littlewood conjecture was refuted when Littlewood (1914) proved that the inequality (1914) proved that the inequality reverses infinitely often for sufficiently reverses infinitely often for sufficiently large n.large n.

Chebyshev put limits on the ratio Chebyshev put limits on the ratio

8

9

)ln(

)(

8

7

nnn

Hadamard's proof of the prime Hadamard's proof of the prime number theorem depends on the number theorem depends on the simple trigonometric inequality simple trigonometric inequality

Distribution of primesDistribution of primes

Wolfram Demonstration Project:Wolfram Demonstration Project: Distribution of PrimesDistribution of Primes Note that Gauss’ approximation Note that Gauss’ approximation

overestimates overestimates (n) for small values (n) for small values of n.of n.

Note also the “pattern” of primes in Note also the “pattern” of primes in the numeric array at the left.the numeric array at the left.

The function (x), the prime-counting function, gives the number of primes less than or equal to x. The plots shown are of the differences between (x) and the approximations to (x) due to Gauss, Legendre, and Riemann.

Zeta (Zeta () Function) Function

““The zeta function is probably the The zeta function is probably the most challenging and mysterious most challenging and mysterious object of modern mathematics, in object of modern mathematics, in spite of its utter simplicity…The main spite of its utter simplicity…The main interest comes from trying to interest comes from trying to improve the Prime Number Theorem, improve the Prime Number Theorem, i.e. getting better estimates for the i.e. getting better estimates for the distribution of the prime numbers.”distribution of the prime numbers.”

Zeta (Zeta () Function ) Function (Summation form)(Summation form)

1

1)(

knk

n

Product Form of Zeta Product Form of Zeta FunctionFunction

kp is the pth prime.

1 11

1)(

kspk

s

Riemann HypothesisRiemann Hypothesis

Prove the Riemann Hypothesis and Prove the Riemann Hypothesis and win a million bucks!win a million bucks!

http://www.claymath.org/http://www.claymath.org/millennium/Riemann_Hypothesis/ millennium/Riemann_Hypothesis/

Peter Sarnak’s descriptionPeter Sarnak’s description

Riemann HypothesisRiemann Hypothesis

Essentially, the Riemann Essentially, the Riemann Hypothesis says that Hypothesis says that everyevery zero of zero of (s) lies on the line x (s) lies on the line x = ½.= ½.

SongSongAnother variation says that Another variation says that (1/2 + it)=0 (1/2 + it)=0

MMööbius Function bius Function (n)(n)

The Zeta function is The Zeta function is connected to the connected to the MMööbius function by bius function by the following the following equation:equation:

primesdistinctkofproductaisnif

nif

factorsrepeatedmoreoronehasnif

nk)1(

11

0

)(

)(

1)(

1 sn

n

ns

Quantum Physics & Riemann Quantum Physics & Riemann ZerosZeros

In a chance meeting over tea at In a chance meeting over tea at Princeton, physicist Freeman Dyson Princeton, physicist Freeman Dyson and number theorist Hugh Montgomery and number theorist Hugh Montgomery happened to talk about what they were happened to talk about what they were each working on. Dyson was looking at each working on. Dyson was looking at the energy levels of electrons in heavy the energy levels of electrons in heavy atoms, Montgomery was looking at the atoms, Montgomery was looking at the zeros of the zeta function. The zeros of the zeta function. The patterns were the same.patterns were the same.

Moments of Moments of on the critical on the critical lineline

At the RHI conference in Seattle in At the RHI conference in Seattle in 1995, some mathematicians and 1995, some mathematicians and physicists said that there is a factor, physicists said that there is a factor, ggkk in a formula for moments of z that in a formula for moments of z that match eigenvalues of atoms.match eigenvalues of atoms.

They knew the first two factors, and They knew the first two factors, and after some work found the third.after some work found the third.

The fourth (and formula) were found The fourth (and formula) were found by 1998.by 1998.

Moments of Moments of on the critical on the critical lineline

gg11=1=1

gg22=2=2

gg33=42=42

gg44=24,024=24,024

1

0

2

)!(

!!k

jk kj

jkg