E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010Lecture 4: Sieves, Spirals and Growth1. ObjectiveIn this worksheet, we get to know prime numbers better. We look at the most primitive algorithmto compute all primes: the Sieve of Erasthostenes. An other way to visualize primes is theUlam spiral. Finally, we look at the growth rate o f the prime numbers.The Sieve of Erasthostenes1) first cross off all the numbers which are larger than 2 and divisible by 2. Then cross off all thenumbers which are larger than 3 and divisible by 3. etc. Continue with p = 5, 7. Which numbersare left?928272625242322212293837363534333231339484746454443424144958575655545352515596867666564636261669787776757473727177988878685848382818899897969594939291991009080706050403020101019181716151413121112) With how many numbers do we have to sieve to make sure that the number 229 is prime?The Ulam Spiral3) Fill in the primes into the Ulam spiral.123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240The growth rateThe following g r aph shows the sequencean=pnn log(n)where pnis the n’th prime number. The prime number theorem theorem assures that the sequenceangoes to 1.200004000060000800001000001.1301.1351.1404) (more difficult) How many primes do we expect on one round on the Ulam spiral far away?E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010Lecture 4: prime numbers1. ObjectiveThis worksheet deals with prime numbers. Euclid proved in his Elements that the number ofprimes is infinite. The same proof works to see that certain classes of primes are infinite. We lookat an other proof which gives more insight and shows that there are arbitrary large gaps in theset of prime numbers.Euclid’s ProofEuclid’s proof starts with the assumption that there are only finitely many primes. Enumeratethem p1, ...., pk. We can now form the number n = p1p2· · · pk+ 1. This number is not divisibleby any of the primes b ecause it leaves rest 1. Since the number n can not be divisible by anyprime, it must be a prime itself, but larger than any pj. This contradicts that the list of primeswas complete.1. Primes of the form 4n + 3.We prove that there are infinitely many primes of the form 4n + 3. We assume that there are onlyfinitely many primes p1, ..., pkof the form 4n + 3.a) Show that 4p1p2...pk− 1 is aga in of the form 4n + 3.b) Can 4p1p2...pk− 1 be a prime?c) Can 4p1p2...pk− 1 have a factor of the f orm 4n + 3?We conclude that all factors of this number are o f the form 4n + 1.2. Arbitrary large gaps of primesFor every n, there exist consecutive primes which differ by at least n.a) Show that all integers n! + 2, ....n! + n are composite.b) Let pkbe the largest of the primes not greater than n! + 1 Then pk+1is larger than n! + n.E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010Lecture 4: Gaussian integers1. ObjectiveWe look at prime numbers in space of Gaussian integers n + im. These are numbers differentfrom the units {1, i, −1, −i} which only have a unit or itself as a factor. In other words, if(n + im) = (a + ib)(c + id), then one of the two factors is a unit and the other is (n + im).The length of a Ga ussian prime is defined as |z| =√a2+ b2. One of the properties of complexmultiplication is that |zw| = |z|·|w|. Note that the length of a Gaussian integer does not need tobe an integer. In order to check that a number n is prime, we have to divide by all number a + ibof length smaller or equal tha n√n.Addition Multiplication Division3 + i + 2 + 5i = 5 + 6i (1+i) (2+i) = 1+3i (1+i)/(2+3i) = (1+i) (2-3i)/(4+9)2. Problems1. Is 5 a Gaussian Prime? Hint: Try a factor p = 2 + i.2. Is 3 a Gaussian Prime? What about 7?3. If a real number is a Gaussian prime, then it is also a regular prime.4. If p is a Gaussian prime. Why are −p, −pi, pi Gaussian primes too?5. Verify that if a regular prime p can not be written a s a2+ b2, then it is a Gaussian prime. [You can assume that if it can be factored, then its factors are of the form a + ib, a − ib. ]6. Any number a2+ b2always leaves the rest 1 when dividing by 4.7. Any prime number which leaves rest 3 when dividing by 4 is a Gaussian prime.8. There are infinitely many Gaussian primes.E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010Lecture 4: Wilson’s theorem1. ObjectiveWilson’s theorem is a simple formula characterizing primes. We look here at why the theorem istrue.2. Wilson’s theoremn is a prime if and only if (n − 1)! + 1 is divisible by n.3. Proof of the theoremThe proof of the theorem has two directions:If n is a prime, then the equation xy = 1 mod n with different x, y has exactly one pair of solution.For x2= 1, there is only the solution 1, −1.123456789101112131415160Wilson’s theorem in the case p = 17. We find all pairs which multiply to 1 Like 2 ∗ 9 = 18, 3 ∗ 6 =18, 4 ∗ 13 = 52, 8 ∗ 15 = 120 which all leave rest 1 when dividing by 17. Only t he numbers1 and −1 do not pair. The product (n − 1)! multiplies all the numbers together and gives(−1) · 1(2 ∗ 9)(3 ∗ 6)(4 ∗ 13)(5 ∗ 7 )(8 ∗ 15)(10 ∗ 12)(11 ∗ 14) = −1.Problem 1) Verify the proof either in the case p = 13 o r p = 29.1234567891011120123456789101112131415161718192021222324252627280If n = pq is not a prime and larger than 4, then (n − 1)! is divisible by n because it is a multipleof p and q.Problem 2) Verify this in the concrete case of n = 15. Why is15! = 1 ∗ 2 ∗ 3 ∗ 4 ∗ 5 ∗ 6 ∗ 7 ∗ 8 ∗ 9 ∗ 10 ∗ 11 ∗ 12 ∗ 13 ∗ 14a multiple of 15?3. Lattice pointsThe Gamma function Γ(x) is a function which is forpositive integers equal to (x − 1)!. It interpolat es thefactorial function.Problem 3) Convince yourself that the primes are inte-ger lattice po ints on the graph of the function f(x) =(Γ(x) + 1)/x, where Γ(x) is the Gamma function. Tothe right, we see the function from 2 t o 7, where therounded value in (−1/2,