E-320: Teaching Math with a Historical Perspective Oliver Knill, 2011Lecture 13: Experimental mathematics1. ObjectiveWe look at some problems in which computers could play a role. Just if a counter example wouldexist:2. Goldbach’s conjectureThe statement is that every even number larger than 2 is the sum of two primes. While it isunlikely that a computer search will find a counter example, it might be that computer searchfinds the solution to the problem. How?3. Euler CuboidIt is not known whether there exists a cuboid with integer side length such that all face diagonalsare integers and additionally, also the lar ge diagonal is an integer. Computer searches are on theway. One can search on parametrized families of Euler cubes.4. Riemann hypothesisThis is a prototype where experimens led to more and more support that the Riemann hypothessis true. One approach looks numerically for roots of the Riemann zeta function on the criticalline. An other approach is the Mertens appro ach. Define µ(n) = 1 if n is the product of an evennumber of different primes and −1 if n is the product of an odd number of different primes. Inall other cases, that is if n has a square factor larg er than 1, we have µ(n) = 0. Is µ sufficientlyrandom so that Sn=Pnk=1µ(k) grows like the iterated law of logarithm? While it looks as if theµ(n) behave like a random sequence, there are some correlations.5. BlliardsOne does not know whether there are triangular billiards without periodic po ints.One also does not know whether there are smooth convex billiards besides the ellipse for whichone has integrability in the sense that all points are either periodic, asymptotic to a periodic pointor almost periodic.6. Chaos in the standard mapWe have seen the problem before. Verify that for c > 2 and all n1nR10R10log |∂xfn(x, y)| dxdy ≥log(c2). The left hand side converges t o t he average of the Lyapunov exponents which is in thiscase also the entropy of the map.5. Are there prime twins?This is a pro blem, which can first of all be investigated statistically. Similarly as Gauss lookednumerically for a law describing the frequency of the prime numbers, one can first see, hoe manyprime twins one has to expect in a certain interval and then see whether this expectation isconfirmed. Furthermore one can look whether there are any patterns on arithmetic subsequences.Maybe there are some unexpected sequences a long which there are more prime twins. Related tothe twine prims problem is the problem to estimate the minimal distance between two Gaussianprimes in the complex plane.5. Is π normal?The decimal digits of π appear random enough so that every digit appears with the same frequency.6. Are there odd perfect numbers?While a brute f orce search is unlikely, there are other approaches which are more likely to find anodd perfect number. Any perfect number satisfies σ(n)/n = 2, where σ(n) is the sum of all thedivisors of n including n. Take a large set B = {p1, ..., ps} of primes. For every k = (k1, ..., ks),form the number n = pk11pk22...pkls. We have σ(pk) = ( 1 + p + p2+ ...pk)/pk= (p − p−k)/(p − 1).Let a(p, k) = log(σ(pk)). The goal is to find (k1, ..., ks) such thatlog(a(p1, k1)) + ... + log(a(pl, kl)) − k1log(p1) − ...kslog(ps) = log(2) .The idea is to keep first the primes and change only the ”dials” kjin a controlled way. Certaindial changes will produce a very small net change of the left hand side. For large primes and largen the first order change will dominate and methods from Diopha ntine geometry could be usedand linear algebra to get close to the right hand side. If lucky (provided of course there is an oddperfect number), one could hit it like this. If we would take 1000 primes each 1000 digits long anddeal with exponents of the order of 1000, we would investigate numbers with billions of digits.Worksheet: Turing machines1. ObjectiveWe see a concrete Turing machine and evolve it a few steps to see what it does.2. The machineThe machine is initially in state 1 and starts with an empty tape.-3 -2 -1 0 1 2 3 · · ·· · ·0 0 0 0 0 0 0· · ·Here is the definition of a machine with three stat es:Input 0State New state Write move to1 2 0 left2 3 1 right3 1 1 leftInput 1State New state Write move to1 2 1 left2 1 0 right3 3 0 right2. Work the machineWe run this machine a few steps and mark the number, where the machine reads the tape. Thisplace will move with time.-3 -2 -1 0 1 2 3 · · · State=1· · ·0 0 0 0 0 0 0· · · State=1· · ·0 0 1 0 0 0 0· · · State=2· · ·0 0 1 1 0 0 0· · · State=3· · ·0 0 1 1 0 0 0· · · State=1· · ·0 0 1 1 0 0 0· · · State=11231122113. A new machineHere is the definition of a new machine with three states:Input 0State New state Write move to1 2 1 right2 3 1 right3 3 0 leftInput 1State New state Write move to1 2 1 left2 1 0 right3 3 0 left2. It’s your turn!Run it!· · ·0 0 0 0 0 0 0· · · State=1· · · · · · State=· · · · · · State=· · · · · · State=Lecture 13: worksheet the integer partitionproblem1. ObjectiveWe look at a problem which is known to be NP-complete. If it could be solved in polynomialtime, all NP problems would be polynomial and P=NP.In other words, if you can design a method, which solves the problem in a manner which ispolynomial in n, you win a Million dollars and you would have solved the most importa nt problemin computer science.2. The problemGiven n positive integers a1, ..., an, divide them up into two subsets, so that the sum of thesenumbers in one set and the sum of numbers in the other set are as close together as possible.3. An exampleAssumeA = {2, 3, 5, 7, 11, 13, 17} .1. Apply the ”cruel school selection” algorithm. The largest two elements are the team leaders ofa team, who alternatively select the best remaining candidate. What do you get?2. Can you do b etter than that ? How well can you divide up the numbers better?Lecture 13: Worksheet factoring integers1. ObjectiveOne does no t know how to factor integers of 200 digits efficiently. Can we find some factors fast?2. Checking a factor of 91) There is a simple method to decide whether a number is divisible by 9. What is it?2) For example: is121212121212121212divisible by 9?3) There is a simple method to decide whether a number is divisible by 11. What is it?4) For example is1212121212121212121212divisible by 11?5) Which factors can be detected by looking at the last decimal digit of a number only?6) The 1,-3,2 method for