How many prime numbers are there?

Transcription

1 CHAPTER I: PRIME NUMBERS Sectio : Types of Primes I the previous sectios, we saw the importace of prime umbers. But how may prime umbers are there? How may primes are there less tha a give umber? Do they follow ay patter? Do they all have the same structure? Are there formulas that yield the primes? These are just a few of the questios about prime umbers that have itrigued mathematicias for ages. Some have bee aswered ad some have t. I this sectio, we ll discuss these questios (ad more) ad aswer those that we ca. How may prime umbers are there? It was Euclid circa 50 BC who first proved that there are ifiitely may primes. It has sice bee proved by others i differet ways, but Euclid s proof is the most elegat, precisely because it is the simplest. Theorem.. (Euclid) There are ifiitely may primes. Proof: Suppose there are fiitely may primes, say p, p, p,, p. Defie the umber N = p p p p +. By the FTA, this umber is divisible by some prime umber. Suppose without loss of geerality that it is p. So p divides N ad it obviously divides p p p p as well. So by Exercise.. (), it must divide also. This is clearly impossible. Therefore our iitial assumptio (that there were oly a fiite umber of primes) is false. So there are ifiitely may primes. Oce a theorem is proved the issue is settled, right? Not quite. Sometimes, people try to fid a differet method of provig a theorem because they are ot satisfied with the method employed previously. I this case, some mathematicias objected (ad i fact some still object) to Euclid s use of proof by cotradictio. I the early part of the 0 th cetury there were three competig schools of thought regardig the ature of mathematics. Oe of them, the ituitioist school of thought, held that proofs by cotradictio were ivalid. This way of thikig was ot ew, but this was the first time their beliefs were orgaized ito a coheret viewpoit. The ituitioists, however, did ot attempt to prove the ifiitude of the primes usig a differet method. For oe reaso, this had already bee doe by Euler (i 748) ad Tchebeychef (i 85). But more importatly, they did ot eve believe that there were a ifiite umber of primes. I fact, ituitioists reject ifiity altogether. Needless to say, their viewpoit did ot become the predomiat viewpoit i mathematics. I 748, the Swiss mathematicia Leohard Euler (707-78) devised a secod famous proof of the ifiitude of the primes. To sketch his proof, we will eed his π fuctio. Euler defied π ( ) to be the umber of primes umbers less tha or equal to. So for example, π (5) =, π (6) =, ad π (7) = 4. Notice that π ( ) is a step fuctio that jumps up by oe uit

2 at each prime umber. Now defie λ( ) = π ( ) i= where meas product (similar to how meas sum) ad p, p, p,, p π ( ) are the primes less tha or equal to. It ca (fairly) easily be show that this product is greater tha or equal to. As, this sum approaches the k harmoic series, ad is therefore diverget (teds to ). Sice Euler s λ fuctio is larger tha this, it must also ted to. Therefore, there are a ifiite umber of primes. While it might seem uecessary (eve a waste of time?) to prove a result that has already bee proved, we shall soo see that the techiques (ad fuctios) Euler used i his proof tured out to be very valuable ad have far reachig implicatios. Fially, i 85, Russia mathematicia Pafuti Tchebeychev (8-894) showed that π ( ) l(l( )) (for > ). Sice l(l( )) as, π ( ) must ted to also. Therefore there are a ifiite umber of primes. (This is a great example of a result i the field of aalytic umber theory.) pi k= How may primes are there less tha a give umber? Ok, so it is settled. There are a ifiite umber of primes. But how may primes are there below,000,000? Puttig it aother way, how do we evaluate Euler s π ( )? The earliest successful method for determiig the umber of prime umbers less tha a give umber was devised by the aciet Greek mathematicia Eratosthees (76-94 BC). Eratosthees was a reaissace ma 700 years before the Europea Reaissace. I additio to beig a woderful mathematicia, he was also a geographer, astroomer, historia, poet, ad athlete. Eve his ickame Beta reflected his prowess at a wide variety of fields. Leged has it he acquired that ickame because he was cosidered to be the secod best i the world at everythig! While this might appear to be a criticism, it would obviously still be quite a achievemet. With regards to umber theory, Eratosthees is best remembered for computig π ( ) (although it was t called this util Euler) usig his prime umber sieve (ow called the Sieve of Eratosthees). It works like this: write dow all the umbers from to. Cross out the umber (sice it is ot prime). Circle the umber ad the cross out every secod umber (every multiple of ). The circle the umber ad cross out every third umber (uless it has already bee crossed out). Circle the ext utouched umber ad cross out all of its multiples. Cotiue util every umber has bee circled or crossed out. The circled umbers are all prime, so just cout them up. Exercise.. Use the Sieve of Eratosthees to fid π (400). While this method works, it is obviously burdesome for large values of. Aother way to approximate how may prime umbers there are less tha or equal to a give umber is the celebrated Prime Number Theorem.

3 Theorem.. (Prime Number Theorem) For large values of, the umber of primes less tha or equal to is approximately equal to. I other words, l( ) π ( ) lim =. l( ) This was first cojectured by Gauss i 800 ad was prove the first time i 896 idepedetly by the Frech mathematicia Jacques Hadamard ad the Belgia Charles de la Vallee-Poussi. Atle Selberg of Norway ad Paul Erdös of Hugary proved it agai i the middle of the 0 th cetury, this time without resortig to complex aalysis as Hadamard ad Vallee-Poussi had. The Prime Number Theorem is oe of the most impressive results of aalytic umber theory. Exercise..4 Use the Prime Number Theorem to approximate how may primes are betwee 07 ad 408 iclusive. dx Gauss actually foud that the defiite itegral l( x ) approximates π ( ) eve better tha. For years, this itegral, amed Li( ), was assumed to be larger tha π ( ) l( ) for all. However, i 94 the Eglish mathematicia J. E. Littlewood proved that this was ot the case, ad i 9 a umber N was foud for which Li( N) < π ( N). As a fa of the power of mathematics, I wat to reiterate that last poit. Mathematicias thought that Li( ) > π ( ) for all, but i 94 Littlewood proved that was ot the case. So Littlewood proved that Li( ) π ( ) was possible. But it took 9 years for someoe to actually fid a umber for which this was true. Now kow as Skewes umber (after the mathematicia who foud it), N = 0 70 loger the smallest such umber. It is ow kow that betwee ad 80 there are 0 itegers for which Li( ) π ( ). Simply amazig This is o Li( ) Exercise..5 Show that lim =. [Hit: Use the Prime Number Theorem, L Hopital s π ( ) Rule, ad the Secod Fudametal Theorem of Calculus] Do they follow ay patter? So we ow kow there are ifiitely may prime umbers, ad we ca roughly approximate how may there are less tha a give iteger. What ca we say about their form or structure? Aythig? Sure. First of all, except for, they are all odd. Stated aother way, except for, they are all of the form a+ for some iteger a. Similarly, except for, they are all of the form a+ or a+. May other such statemets usig differet base primes ca be made. These

4 statemets are special cases of a woderful theorem established by Peter Dirichlet ( ) of Germay. Theorem..6 (Dirichlet) If a ad b are relatively prime, the i ay arithmetic progressio a, a+ b, a+ b, a+ b,, there are a ifiite umber of primes. It is however, impossible that such a arithmetic sequece be made up etirely of primes. Exercise..7 (a) I Drichlet s Theorem, why must a ad b be relatively prime? (b) Why is it impossible for a arithmetic progressio as described i Dirichlet s Theorem to be made up etirely of prime umbers? Ok, so a arithmetic sequece caot be oly prime umbers. Suppose you wat a arithmetic sequece i which the first terms are all primes. Is this possible for ay? This is ukow. The logest such strig is curretly achieved by the arithmetic progressio k for k = 0,,,. I searchig for patters amog the primes, arithmetic progressios (liear fuctios) are oly the begiig. We ca also cosider quadratic fuctios. The search for prime-producig quadratic fuctios is very iterestig. Cosider the simple quadratic polyomials f p( ) = + + p. Example..8 (a) Let p=. The f = + +. This polyomial, whe evaluated at ( ) itegers startig with = 0 yields prime umbers. Namely, f (0) = ad f () = 5. (b) Let p= 5. The f = + +. This time we get 4 iitial prime umbers, ( ) 5 5 f (0) = 5, f () = 7, f () =, ad f 5 () = 7. (c) Let p=. The umbers. f = + +. The first evaluatios produce prime ( ) See a patter? It appears that the quadratic polyomial f p( ) = + + p will always yield p iitial prime umbers whe evaluated at the itegers 0,,, p. Could we ever get more? Will it ever be less? No, ad yes. Notice that f p ( p ) ( p ) ( p ) p p p p p p = + + = =, so f ( p ) will ever be prime, ad therefore we will ever get p iitial primes. Also, it has p bee show that p= 4 is the largest prime umber for which we will get p iitial primes. For example, f 4 yields oly oe iitial prime (sice f 4 () = 45 ).

5 Aother way i which mathematicias have searched for a patter is i lookig at the distributio of the primes. There are very small itervals betwee cosecutive primes (such as ad, 4 ad 4, ad,000,000,000,06 ad,000,000,000,06) as well as arbitrarily log itervals. Primes i the former case are called twi primes. It is a uaswered questio whether there are ifiitely may pairs of twi primes. Cosecutive primes ca also be very far apart. I fact, it is easy to show that the umbers ( + )! +,( + )! +,,( + )! + ( + ) (for some fixed atural umber ) are all composite. So if we wish to fid 5 cosecutive composite umbers (ad therefore a gap i the primes of at least that big), we eed oly look at the umbers, 6! + = 7 6! + = 7 6! + 4= 74 6! + 5= 75 6! + 6= 76 Of course, this may ot be the first occurrece of such a gap. I fact, 90, 9, 9, 9, 94, 95, ad 96 are all composite. Aother curret hot topic i prime umber theory (utilizig the computer i o small amout) is lookig for the first occurrece of a gap for each possible size. The largest effectively calculated gap has legth 55 ad it occurs first after the prime 40,49,95,999,5,707. Exercise..9 Fid 0 cosecutive composite itegers. This leads us to Goldbach s Cojecture. I a letter to Euler i 74, Christia Goldbach guessed that every eve iteger greater tha 4 is the sum of two primes. This has bee verified for may umbers, but a proof remais urealized. Exercise..0 Assumig Goldbach s cojecture is true, prove that there exists a way to write every iteger greater tha as the sum of at most three primes. Do they all have the same structure? Certaily ot. I fact, primes come i a surprisigly wide variety of structures. Throughout history, primes of certai types have bee studied for oe reaso or aother. They are ofte amed for the primary mathematicia who studied them. There are Fermat primes (prime of the form + studied by Pierre Fermat), Germai primes ( p studied by Sophie Germai), ad Mersee primes ( p studied by Mari Mersee). I the first formula, is a atural umber; i the secod two p is a prime. I some cases there are ifiitely may primes (Germai) while i others there are oly fiitely may (there are curretly 47 kow Mersee primes ad 5 kow Fermat primes). We will look ito some of these i more detail later. Exercise.. Fid the first 5 Fermat primes, Germai primes, ad Mersee primes.

6 Exercise.. Show that if a is prime, the a= ad must be prime. [Hit: x = ( x )( x + x + + x+ ) ] Are there formulas that yield the primes? Yes, but ot very hady oes. Aside from the limited formulas give above by f p( ) = + + p, there are a few formulas that produce primes. Most of these formulas are very u-utilitaria however. I oe (see Exercise.. below), i order to fid, say the 0 th 0 prime, you must cout the umber of primes less tha or equal to. There are 7 primes less 0 tha or equal to, but i coutig this, we ve goe right past the oe we wated to fid. There is a eve more amazig, yet useless formula for primes. I 957, America mathematicia William Mills proved that there exists a positive real umber r such that f ( ) = [ r ] is prime for ay atural umber (where the symbol [ ] deotes the greatest iteger fuctio). However, o oe, icludig Mills, kows what that real umber r is. This is a perfect example of a existece theorem. Fially, there is also a amazig polyomial that produces every prime umber. It is a 5 th degree polyomial i 6 ukows, ad whe evaluated at positive itegers, if a positive iteger is achieved, it s prime. Exercise.. I 964, Williams devised the followig formula: p = + m= + π ( m). Agai, [ ] deotes the greatest iteger fuctio, ad π ( m) is of course Euler s fuctio. Use this formula to fid the 5 th prime.