Additional comments (Math 313) related to material from the class. These additional remarks are for your enjoyment, and will not be on homeworks or exams. These are just meant to suggest additional topics worth considering, and I am happy to discuss any of these further.

• Friday, May 12. Video: Dr Seuss' Circle Method: https://youtu.be/vDxCIkCg2ik
  • The goal today was to give a sense of the circle method, of how we can write numbers as sums of primes.
  • https://en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood_circle_method (email me if you want the chapters from my book)
  • Dr Seuss attempt
    • Drawn from the following:
      • http://www.mfwi.edu/MFWI/Recordings/One%20Fish.pdf
      • http://denuccio.net/ohplaces.html
      • http://teacherweb.com/VA/WarwickHS/Elliot_T/Text-The-Sneetches.pdf
      • http://paulandlizdavies.com/poems/cat.htm

    • I stood there with evens
      I stood with primes too
      And I said how I wish
      I could get all with just two  
      
      The primes aren't so dense, they are really quite small
      You might think such a set wouldn't be dense enough for all
      
      We need an approach to prove what is expected
      Fortunately Hardy and Littlewood gave us their Circle Method
      
      The method will help us. It has what we need.
      And its idea is quite simple. And it works at great speed.
      And with four primes it's one hundred per cent guaranteed!
      
      We will pick up the primes, you will see something new
      Two terms, and we call them Term One and Term Two.
      
      At first you won't know just what you need to do
      To ensure that Term 1 is much larger than Term 2
      
      The error term shouldn't contribute, we must make it small
      It should not be seeable as the main term's almost all
      
      the first term's a good term, it is tame.  oh, so tame!
      it shows right here what we expect to be true 
      that every even number is a sum of a prime 1 and prime 2
      
      here is the game that you should all like
      we approximate the value by looking at small spikes
      
      there that sum runs about
      with big bumps, jumps and ticks
      and with a little care
      and all kinds of good tricks.
      we do like the way that issues go away
      if only term 2 were like this,
      oh, what would we say!'
      
      we look up and down term 2's sum, we look 'em over with care.
      We try many approaches, but none are good enough to compare
      Out there things can happen
      and frequently do
      to sums as variable
      and wild as term 2.
      
      The sum just taunts us as we try our approaches
      After each attempt all that's left are reproaches
      We never have learned
      It's so big it encroaches
      
      But these fears are quite wrong. I'm quite happy to say
      If you're willing to modify what we shall prove on this day
      If instead its decided that we use three primes not two
      And work with odd numbers instead of well you-know-who
      If we decide that it won't matter how many summands
      That having four primes is as good as just two, and
      In that case, we can forget about all the errars
      And whether each is a sum of two, three or four of thars.
      

   
• Wednesday, May 10. Video: Polynomial Fermat's Last Theorem: https://youtu.be/2jzReEvf7eA
  • Polynomial Fermat's Last Theorem: in addition to the book see https://www.parabola.unsw.edu.au/files/articles/1990-1999/volume-35-1999/issue-1/vol35_no1_1.pdf and for difficulties in going from the polynomial case to the integer case see https://math.stackexchange.com/questions/1618793/why-doesnt-fermats-last-theorem-for-polynomials-entail-that-for-integers
  • Polynomial abc: https://en.wikipedia.org/wiki/Mason%E2%80%93Stothers_theorem and http://www.math.purdue.edu/~egoins/seminar/12-09-14.pdf
  • Fermat for fractional exponents: https://math.williams.edu/fermats-last-theorem-for-fractional-exponents/ and http://www.maa.org/sites/default/files/pdf/cmj_ftp/CMJ/May%202010/3%20Articles/1%20Morgan/Morgan9_5_09.pdf
  • The main theme of today's lecture was on how to attack problems. What conditions should we add? How do we figure out what restrictions to add?
   
• Monday, May 8. Video: Elliptic Curve Cryptography and Integer Points: https://youtu.be/M-bRTl4H_KQ
  • A big part of today is emphasizing how we can solve problems by brute force algebra (as you noticed in your reading), but such an approach often leaves us unsure of the why. Here for example is a high level proof of associativity of the group law: http://people.reed.edu/~jerry/311/rr.pdf . I want you to see that you can approach things by doing algebra, but that at the end of the day this is not always the best approach! It is of course great to know something is true, but we often want the why as well. The semester is ending, you're busy, we worked hard earlier in the year. These classes as springboards. Think of yourselves as Newtons with access to Kepler's data; find the why!
  • http://wstein.org/simuw06/ch6.pdf (has elliptic curve cryptography)
  • Elliptic curve Diffie-Hellman: https://en.wikipedia.org/wiki/Elliptic_curve_Diffie%E2%80%93Hellman
    • Difference with RSA version: https://security.stackexchange.com/questions/46802/what-is-the-difference-between-dhe-and-ecdh
  • Counting points on an elliptic curve:
    • https://en.wikipedia.org/wiki/Counting_points_on_elliptic_curves
    • https://en.wikipedia.org/wiki/Nagell%E2%80%93Lutz_theorem
      • http://www.uio.no/studier/emner/matnat/math/MAT4250/h14/ell2.pdf

• Friday, May 5. Video: Elliptic Curve Group Law and the Congruent Number Problem: https://youtu.be/RqJfLqyWMFM
  • Proof of elliptic curve group law: http://math.rice.edu/~friedl/papers/AAELLIPTIC.PDF (see also http://www.math.ku.dk/~kiming/lecture_notes/2000-2001-elliptic_curves/grouplaw.pdf)
  • Congruent number problem: https://en.wikipedia.org/wiki/Congruent_number
    • Notes by Keith Conrad: http://www.math.uconn.edu/~kconrad/blurbs/ugradnumthy/congnumber.pdf
  • A great example of what is locally best is not globally best are the two papers on whether or not dogs (or one dog in particular, Elvis) know calculus. Looking at the the path the dog takes to get the stick in the water, if the dog is in the water close to the stick the optimal choice is to swim straight for the stick; however, if the stick is far away it's best to swim to water, run on water, then jump back in. In other words, if you only can go for a few seconds the closest you can get to the stick is to swim for it; however, if you're in it for the long haul it's worthwhile to do something which is not locally optimal but which will lead to optimal. For interesting articles related to this, see the two papers below by Pennings on whether or not dogs know calculus. Click here for a picture of his dog, Elvis, who does know calculus.
    • Do dogs know calculus?  
    • Do dogs know bifurcations? This paper discusses bifurcations. An additional aside is that the bifurcation in the dog paper leads to a nice proof of the arithmetic / geometric mean inequality, one of the more important ones in math. For other proofs, seehttp://www.williams.edu/go/math/sjmiller/public_html/OSUClasses/487/ArithMeanGeoMean.pdf.) This is a great example of a phase transition.

• Wednesday, May 3. Video: Discriminants, Elliptic Curve, Points at Infinity: https://youtu.be/MPDxH0Qor4M
  • The mistake in class today in guessing the formula of the discriminant of a cubic was really good, as it highlights that when there are two approaches one could be better than another. We found it easier to work with f(x) = (x-1)(x-2)(x+3) than f(x) = x^3 - 1 or x^3 + 1 to compute the discriminant, and if we know the A dependence then we can use it to get B easily. Unfortunately we made a mistake in A (lost a sign) and that led to a mistake in the coefficient of B. It's better to look at x^3 + 1 or x^3 - 1 as that is independent of the coefficient of A. Thus our mistake shows that it is better to take independent paths, so an error does not propagage -- a very valuable lesson! (I might deliberately make that mistake next time I teach it.)
  • https://en.wikipedia.org/wiki/Discriminant
  • https://en.wikipedia.org/wiki/Stereographic_projection
  • https://en.wikipedia.org/wiki/Point_at_infinity
  • https://en.wikipedia.org/wiki/Alexandroff_extension
  • https://en.wikipedia.org/wiki/Projective_plane

• Monday, May 1. Video: Introduction to Elliptic Curves, Roots of Polynomials, Discriminants: https://youtu.be/KerQfFu4WEY  (code and pdf of code)
  • https://en.wikipedia.org/wiki/Elliptic_curve
  • https://en.wikipedia.org/wiki/Discriminant
   
• Friday, April 28. Video: Coding and Conjecturing for 3x+1: https://youtu.be/nwEdbWXc8AQ
  • Deliberately not having any comments today as don't want to color your conjecturing.
  • Watch later: Numberphile on 3x+1: https://www.youtube.com/watch?v=LqKpkdRRLZw&feature=youtu.be  
   
• Wednesday, April 26. Video: 3x+1 Problem: https://youtu.be/RcHmAbtfL7g (mathematica code and pdf) (unfortunately video did not project, have to look at code)
  • Deliberately not having any comments today as don't want to color your conjecturing.
  • Watch later: Numberphile on 3x+1: https://www.youtube.com/watch?v=LqKpkdRRLZw&feature=youtu.be
   
• Wednesday, April 19. Video: Hensel's lemma: https://youtu.be/zlsAbRTxmWQ
  • Hensel's lemma: https://en.wikipedia.org/wiki/Hensel%27s_lemma
  • p-adics: https://en.wikipedia.org/wiki/P-adic_number

• Monday, April 17. Video: Sniffing out eqns, Series Expansions for DiffEQs, Pascal's Triangle, Steganography, Solving mod prime powers https://youtu.be/eRuKhHYJpTg
  • https://www.youtube.com/watch?v=tt4_4YajqRM (my movie constructing table mod 2)
  • https://www.youtube.com/watch?v=QBTiqiIiRpQ (mod different primes)
  • Hensel's lemma: https://en.wikipedia.org/wiki/Hensel%27s_lemma
  • Power series solutions of differential equations: https://en.wikipedia.org/wiki/Power_series_solution_of_differential_equations
  • Steganography: https://en.wikipedia.org/wiki/Steganography
  • Pascal's triangle mod 2: http://ecademy.agnesscott.edu/~lriddle/ifs/siertri/pascal.htm
   
• Friday, April 14. Video: Method of Descent for Fermat and Introduction to p-adics: https://youtu.be/DIwYrmIxQpg
  • Method of Descent: https://en.wikipedia.org/wiki/Proof_by_infinite_descent
  • Strange sums: https://www.youtube.com/watch?v=w-I6XTVZXww    https://www.youtube.com/watch?v=PCu_BNNI5x4 
  • Fun video on Euler's constant: https://www.youtube.com/watch?v=4k1jegU4Wb4
  • p-adics: https://en.wikipedia.org/wiki/P-adic_number
   
• Wednesday, April 12. Video: Introduction to Diophantine Equations, the Pythagorean Theorem and Pythagorean Triples: https://youtu.be/GW3zoVUZ4Xs
  • Diophantine equations: https://en.wikipedia.org/wiki/Diophantine_equation
  • Pythaorean triples: https://en.wikipedia.org/wiki/Pythagorean_triple
  • The Pythagorean Theorem relates the length of the hypotenuse of a right triangle to the lengths of the sides (President Garfield is credited with a proof). For us, this result is important as gives us a way to compute the length of vectors.
  • Garfield, Williams alum and US President, has a proof of this; click here for his proof. The proof we used was via dimensional analysis, a wonderful and important technique. The dimensional analysis proof is here.
  • Here is a talk I gave on dimensional analysis and proving and extending the Pythagorean formula:
    • Extending the Pythagorean Formula: http://youtu.be/idIHcgapMG4 (slides here)
      • Modified version (all slides, no blackboard): http://youtu.be/8HtABCG6ACg  (slides here)
      • Expanded version (all slides, 50 minutes):  (slides here)

   

• Monday, April 10: Video: Computing Continued Fraction Expansions: https://youtu.be/eKFrbmIOlq8

  • Silver ratio:https://en.wikipedia.org/wiki/Silver_ratio
  • Metallic mean: https://en.wikipedia.org/wiki/Metallic_mean (generalization of silver and golden mean)
  • One nice expression for \(\pi\) is \(\frac4{\pi} \ \ = \ \ 1 \ + \ \cfrac{1^2}{2\ +\ \cfrac{3^2}{2\ +\ \cfrac{5^2}{2\ +\ \cdots}}}\ .\) There are many different types of non-simple
    expansions, leading to some of the most beautiful formulas in mathematics. For example, \( e \ = \ 2 \ + \ \cfrac{1}{1\ + \ \cfrac{1}{2\ + \ \cfrac{2}{3 \ + \ \cfrac{3}{4 \ + \ \cdots}}}}\ . \)  For some nice articles and simple and non-simple continued fraction expansions, see https://crypto.stanford.edu/pbc/notes/contfrac/nonsimple.html.
  • We saw how nicely we can compute continued fraction expansions of numbers. In particular, for numbers such as \(e\) and \(\pi\) we keep having to find the largest integer less than a ratio of two linear expressions in the quantity of interest.
  • Finding continued fraction expansions of algebraic numbers:
    • https://www.jstor.org/stable/2005108?seq=1#page_scan_tab_contents
    • http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.135.107&rep=rep1&type=pdf
  • Waring's problem: https://en.wikipedia.org/wiki/Waring%27s_problem    
    • sums of two squares: https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares

   

• Friday, April 7: Video: Transcendental Numbers and Continued Fractions: https://youtu.be/kPDSyW6CL9Q

  • Liouville's number: https://en.wikipedia.org/wiki/Liouville_number
  • Normal numbers: Most numbers are normal (https://en.wikipedia.org/wiki/Normal_number), but similar to transcendental numbers it is hard to find a specific example. One is Champernowne's constant: https://en.wikipedia.org/wiki/Champernowne_constant
  • Continued fractions: https://en.wikipedia.org/wiki/Continued_fraction
    • One nice expression for \(\pi\) is \(\frac4{\pi} \ \ = \ \ 1 \ + \ \cfrac{1^2}{2\ +\ \cfrac{3^2}{2\ +\ \cfrac{5^2}{2\ +\ \cdots}}}\ .\) There are many different types of non-simple
      expansions, leading to some of the most beautiful formulas in mathematics. For example, \( e \ = \ 2 \ + \ \cfrac{1}{1\ + \ \cfrac{1}{2\ + \ \cfrac{2}{3 \ + \ \cfrac{3}{4 \ + \ \cdots}}}}\ . \)  For some nice articles and simple and non-simple continued fraction expansions, see https://crypto.stanford.edu/pbc/notes/contfrac/nonsimple.html.

   

• Wednesday, April 5: Video: Liouville's Theorem: https://youtu.be/h4IJNV_mjEA

  • algebraic numbers: https://en.wikipedia.org/wiki/Algebraic_number
  • transcendental numbers: https://en.wikipedia.org/wiki/Transcendental_number
  • Number field: https://en.wikipedia.org/wiki/Algebraic_number_field
    • careful: unique factorization can fail: https://en.wikipedia.org/wiki/Algebraic_number_field#Unique_factorization_and_class_number
    • nice paper with example from Hale Trotter: https://www.jstor.org/stable/2323570?seq=1#page_scan_tab_contents
    • another note: https://rjlipton.wordpress.com/2016/01/30/failure-of-unique-factorization/
  • Liouville's theorem: https://en.wikipedia.org/wiki/Liouville_number#Liouville_numbers_and_transcendence
    • key ingredient: Mean Value Theorem: https://en.wikipedia.org/wiki/Mean_value_theorem
    • We bypassed a lot of results on minimal polynomials: https://en.wikipedia.org/wiki/Minimal_polynomial_(field_theory)
  • Lecture proving the Fundamental Theorem of Calculus through the Mean Value Theorem: http://www.youtube.com/watch?v=Q1TQtH6POyI (52 minutes)
  • Lecture proving Green's Theorem in a day: https://www.youtube.com/watch?v=Iq-Og1GAtOQ (51 minutes)

  Monday, April 3: Video: Irrationality I: Sqrt(2), Dirichlet's Theorem, Pigeonhole Principle: https://youtu.be/JJJSNHhtMSg

  • Rational irrational proofs (with David Montague), Mathematics Magazine (85 (2012), no. 2, 110--114).  pdf (expanded version: pdf;    letter from Nerode and Tennenbaum on history of proof: pdf)
    • See also http://www.cut-the-knot.org/proofs/GraphicalSqRoots.shtml
  • Irrationality of sqrt(2):
    • standard proof: https://www.math.utah.edu/~pa/math/q1.html
    • geometric proof: https://jeremykun.com/2011/08/14/the-square-root-of-2-is-irrational-geometric-proof/
  • Pigeon hole principle: Lots of great links for readings and problems.
    • http://math.stackexchange.com/questions/194312/show-me-some-pigeonhole-problems
    • http://www.math.lsa.umich.edu/~hderksen/ProblemSolving/PS7.pdf
    • http://artofproblemsolving.com/wiki/index.php/Pigeonhole_Principle
    • http://mindyourdecisions.com/blog/2008/11/25/16-fun-applications-of-the-pigeonhole-principle/
    • http://www.math.ucla.edu/~radko/circles/lib/data/Handout-123-153.pdf
  • Dirichlet's approximation theorem: https://en.wikipedia.org/wiki/Dirichlet's_approximation_theorem (if \(\alpha\) irrational then there exists integers such that \(|\alpha - p/q| < 1/q^2\).
  • Hurwitz' theorem (improves bound to \(1 / \sqrt{5} q^2\):  https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(number_theory)
  • algebraic numbers: https://en.wikipedia.org/wiki/Algebraic_number
  • transcendental numbers: https://en.wikipedia.org/wiki/Transcendental_number
  • Roth's theorem: https://en.wikipedia.org/wiki/Roth%27s_theorem
  • Liouville's theorem: https://en.wikipedia.org/wiki/Liouville_number#Liouville_numbers_and_transcendence

   

• Friday, March 17: Video: Introduction to Generating Functions: Fibonaccis and Cookie Problem: https://youtu.be/lJIwL2tsCXE

  • Generating functions: we discussed how to solve the Cookie Problem earlier in the semester. We found the solution through a combination of investigating small cases and cleverly noticing patterns. Generating functions provide a very powerful machinery to attack problems like this. It's very similar to factoring. We can factor \(x^2 - 7x + 12\) by direct inspection, but for \(x^2 - 6x + 11\) we have to resort to the quadratic formula. It's not too bad, and it's very mechanical so it's easy to use....
  • Generating functions: There are many ways to prove Binet's formula for an explicit, closed form expression for the n-th Fibonacci number. One is through divine inspiration, the second through generating functions and partial fractions. Generating functions occur in a variety of problems; there are many applications near and dear to me in number theory (such as attacking the Goldbach or Twin Prime Problem via the Circle Method). The great utility of Binet's formula is we can jump to any Fibonacci number without having to compute all the intermediate ones. Even though it might be hard to work with such large numbers, we can jump to the trillionth (and if we take logarithms then we can specify it quite well).
    • Generating Functions Handout: This is from a book I'm writing on probability. The first section is motivation, feel free to skim. Section 19.2 is the most important. Section 19.3  is more technical and included for completeness; we won't cover. Section 19.4 talks about convolutions -- you'll find the very beginning if you ever wish to analyze the Catalan numbers.
  • The Circle Method: https://en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood_circle_method
    • Read classics: Hardy and Littlewood: Partitio Numerorum III: http://fuchs-braun.com/media/8cdd73c813c342f8ffff80d1fffffff0.pdf
    • We discussed two of the main ingredients in the Circle Method: (1) writing expressions as Main Term + Error with good control on the error, and (2) proving something happens at least once by showing it happens many times.
      • Notes: https://web.stanford.edu/~ebwarner/CircleMethod.pdf
      • Notes: https://web.williams.edu/Mathematics/sjmiller/public_html/OSUClasses/487/circlemethod.pdf (these are old notes of mine, which were eventually incorporated into a book I wrote).

   

• Wednesday, March 15: Video: Perfect Numbers, Mersenne Primes, Fermat Primes: https://youtu.be/5CtFP-5dxyU

  • Perfect numbers: First few: 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216: See https://oeis.org/A000396
    • Big part of Euclid's theorem are prime numbers of the form \(2^p - 1\), which are known as Mersenne primes
    • Another special sequence: Fermat primes: https://en.wikipedia.org/wiki/Fermat_number :    3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, … (sequence A000215 in the OEIS). These have a WONDERFUL applications in constructable regular n-gons:  https://en.wikipedia.org/wiki/Constructible_polygon
    • See this subpage for heuristic arguments on the density of Fermat primes (gives several, one indicating only finitely many expected, another that there should be infinitely many!): https://en.wikipedia.org/wiki/Fermat_number#Heuristic_arguments_for_density

• Monday, March 13: Video:

  • Average order of an arithmetic function
  • Integral test
  • Euler-Maclaurin Summation Formula
  • Harmonic numbers
  • Special Values of L-functions
  • Gregory-Leibniz formula
  • Prime Number Theorem: we spent a lot of time exploring how we could use this to estimate a lot of quantities we care about. For example, we saw that \(\pi(x) \approx x/\log x\) implies that \(p_n \approx n \log n\) and if the primorial \(p_m# = n\) then \(p_m \approx \log n\) so \(m \approx n / \log n\). We used this to get a good sense of the size of \(\phi(n)\), in particular how small it can be. 
  • Dirichlet's Theorem: see https://www.jstor.org/stable/pdf/2322510.pdf

   

• Friday, March 10: Video: Euler Totient Function: https://youtu.be/tFnemsbV5xw

  • Totient function: We spent a lot of time going over how to find proofs; this is far more important than actually proving results. As you should see from reading the book, there are many ways to prove a result. How do you find them? For the first results we went back to the Chinese Remainder Theorem to explcitly find a representation, and we used the fact that two finite sets are the same size if there is a 1-1 map from one set to another.
  • Our second problem had a proof used Telescoping Series: these are incredibly powerful, and took awhile to see we should view \(\sum_{d|n} \phi(d)\) as \prod_{\ell=1}^k \left(\phi(1) + \phi(p_\ell) + \cdots + \phi(p_\ell^{r_\ell})\right)\) where \(n = p_1^{r_1} \cdots p_k^{r_k}\).
  • For telescoping series and the Fundamental Theorem of Calculus see the March 19th comments from my Calc III class a few years back: http://web.williams.edu/Mathematics/sjmiller/public_html/150/addcomments.htm

   

• Wednesday, March 6: Video: Mobius function, Mobius Inversion, Dirichlet Convolution, Totient Function: https://youtu.be/dvGxcVfyUdY

  • Mobius function (see Mertens conjecture for connections to the Riemann Hypothesis).
  • Totient function
  • Dirichlet convolution (to combine arithmetic functions): this is what we talked about with \(\mu \ast 1 = {\rm Id}\) and the like. If you enjoy algebra I strongly encourage you to read this article.
  • Average order of an arithmetic function

   

• Friday, March 3: Video: Diffie-Hellman, UPC codes, RSA, Euler Totient: https://youtu.be/LW3R1V7GTtM

  • Diffie-Hellman Key Exchange: sharing a secret in public
  • RSA: https://en.wikipedia.org/wiki/RSA_(cryptosystem)

    • History of RSA: http://www.rsa.com/rsalabs/node.asp?id=3120

    • 20 years of attacks on RSA:  Dan Boneh, Notices of the AMS, February 1999, 203-213: http://web.williams.edu/Mathematics/sjmiller/public_html/crypto/handouts/Boneh_TwentyYrsAttacksOnRSA.pdf

    • RSA factoring challenge: http://en.wikipedia.org/wiki/RSA_Factoring_Challenge

  • Digital signatures

  • Bar and related codes:

    • Bar codes

    • Check digit schemes

    • Universal Product Code (25th anniversary article)

   

• Wednesday, March 1: Video: Introduction to Cryptography, Caesar Cipher, Affine Cipher, Permutation Cipher, Vigenere Cipher, Midway: https://youtu.be/0kKow_0HHHQ

  • The big math was Euler's totient function, which is a multiplicative function but not completely multiplicative function.
  • The Caesar cipher:
  • Frequency analysis
  • Kruskal's algorithm
  • The Battle of Midway: My favorite part is how we figured out the target was Midway (click this link).
  • Battle of Midway
  • Operation Fortitude
  • Enigma

  • Here are some good handouts on the Enigma / Ultra, as well as WWII:

    • http://www.nsa.gov/about/cryptologic_heritage/center_crypt_history/publications/wwii.shtml

      • Solving the Enigma: http://www.nsa.gov/about/_files/cryptologic_heritage/publications/wwii/solving_enigma.pdf

      • The mathematics of the Enigma: http://www.nsa.gov/about/_files/cryptologic_heritage/publications/wwii/engima_cryptographic_mathematics.pdf

      • Midway: http://www.nsa.gov/about/cryptologic_heritage/center_crypt_history/publications/battle_midway.shtml

      • How mathematicians helped win the war: http://www.nsa.gov/about/cryptologic_heritage/center_crypt_history/publications/how_math_helped_win.shtml

    • Finally, here are some pics of me with an Enigma machine:

    •   

  • Decrypting links:

    • http://rainbow.arch.scriptmania.com/tools/word_counter.html

    • http://letterfrequency.org    (and also http://letterfrequency.org/#english-language-letter-frequency)

  • Story on the message recently found from the civil war: http://newsfeed.time.com/2011/01/01/better-late-than-never-civil-war-message-decoded/
    • Decrypting it: https://securingtomorrow.mcafee.com/executive-perspectives/decrypting-civil-war-messages/
    • Commentary: https://www.schneier.com/blog/archives/2010/12/civil_war_messa.html

  • Some readings

    • Notes on abstract algebra and group theory and RSA cryptography (from my book An Invitation to Modern Number Theory). 

    • Markov chains and cryptography: Persi Diaconis, The Markov Chain Monte Carlo Revolution. (2008). Bull. Amer. Math. Soc. Nov. 2008. 

    • Some history of RSA.

    • Two centuries on, a cryptologist cracks a presidential code: Rachel Emma Silverman, Wall Street Journal, July 2, 2009.

    • 20 years of attacks on RSA: Dan Boneh, Notices of the AMS, February 1999, 203-213.

    • Group Theory in Cryptography: Blackburn, Cid and Mullan.

    • Braid Group Cryptography: David Garber

    • Elliptic curve Cryptography:

      • Expander graphs based on GRH with an application to elliptic curve cryptography: Jao, Miller (not me!) and Venkatesan

      • Do all elliptic curves have the same difficulty of discrete log?: Jao, Miller (not me!) and Venkatesan

    • Quantum Cryptography:

      • Quantum Cryptography: From theory to practice: Ma (PHD dissertation, University of Toronto)

      • http://www.nbcnews.com/technology/technolog/speedy-boson-machine-could-bridge-classic-quantum-computing-1C7662777

    • General articles from the NSA/CSS historical publications.

   

• Monday, February 27: Video: Legendre Symbols, Elliptic Curve Motivation, Law of Quadratic Reciprocity, Euler's Criterion: https://youtu.be/cBxqW2OkpWM

  • We motivated Legendre symbols and quadratic reciprocity by circling back to the first real homework problem. It turns out that's an example of an elliptic curve. There is an incredible and rich structure there which can be of great use in cryptography. The group structure is more involved than what we have with \((Z/pq\Z)^\ast\), which is good for security!
  • Also note that so many of our proofs today are crucially using the main result of this week's homework, namely that \((Z/p\Z)^\ast\) is a cyclic group generated by an element of order \(p-1\). I like doing some things differently from the book so you can see different approaches, always happy to chat about either. It's some work proving cyclicity, but as it yields so many results so easily, and gives us a good sense of what's happening, I think it's worth it.
  • Legendre symbol
  • Jacobi symbol
  • Euler's criterion
  • Quadratic Reciprocity and Biquadratic (or quartic) reciprocity
  • Elliptic curves
    • Group law
    • Hasse's theorem: we gave a heuristic that the number of solutions modulo p is p; the difference between the conjecture and the truth is always at most \(2 \sqrt{p}\).
  • Primitive roots

• Friday, February 24: Dirichlet's Theorem for Primes in Arithmetic Progression, Number Theory Motivation: https://youtu.be/zG185Ef1gPM

  • Dirichlet's Theorem
  • Structure Theorem for finite abelian groups (enormous overkill for what we need!)
  • Dirichlet characters
  • Dirichlet L-function
  • Complex roots of unity
  • The following would take us far too far afield (what we did today was just a little afield, and worthwhile motivation), but it might be covered in an algebraic number theory class: Dirichlet's class number formula. It's well-worth reading this article to see where the quadratic character (and its generalization, the Jacobi symbol) and quadratic reciprocity go.

   

• Wednesday, February 22 Video: Solving the 2x2x2 Rubik's Cube: https://youtu.be/AteP1iFbVmo

  • Solving the Rubik's cube (thanks Alan Chang and Umang Varma!):
    • 2x2x2: https://www.youtube.com/watch?v=PKZ7pxFyYu0
    • 3x3x3: https://www.youtube.com/watch?v=FO1kOU-3Blw
    • 4x4x4: parity issue: https://www.youtube.com/watch?v=nC4jsBD1L1w
  • Rubik's cube (and the G-d number), and homepage:https://www.rubiks.com/ 

   

• Monday, February 20 Video: Euler's generalization of FlT, Solving polynomial equations, Chinese Remainder Theorem: https://youtu.be/WxpNtlsYbzM

  • Saw can use Fermat's little Theorem to create a primality test, but unfortunately there are issues. While if it says `composite' we can trust it, it cannot distinguish between primes and Carmichael numbers. There are a lot of questions one can ask about these troublesome numbers: what is their structure like, are there infinitely many, is there an easy test for them? Many of these are known, or partially conjectured.
  • For the Fermat's little Theorem primality test to be useful we need to compute quantities such as \(a^{n-1}\) modulo \(n\) very quickly for \(n\) having at least 200 digits. While at first it may seem unreasonable to expect to be able to efficiently compute the \(10^{200}\)th power of a number, there is a well-named method (fast multiplication) that allows us to do so. It's also called the Method of Repeated Squaring or fast exponentiation, and the ability to do this (both for matrices as well as regular numbers) is extremely important.
    • For example, consider the task of computing \(x^100\). One's first instinct is to say we need 100 (or 99) multiplications to evaluate x^100, but it is possible to do this in just 8: x*x, x^2 * x^2, x^4 * x^4, x^8 * x^8, x^16 * x^16, x^32 * x^32, x^64 * x^32 * x^4. The key observation is using the base 2 expansion of 100; this idea is one of the reasons RSA encryption is feasible. For more details, see Chapter 1 of my book,http://press.princeton.edu/chapters/s8220.pdf (especially Sections 1.1 and 1.2.1). Quite often in mathematics we have algorithms to solve problems that are not feasible in practice, and finding efficient ways of computing quantities is a big (and important) industry.
    • We've already seen another great example of where we know the solution exists but have trouble finding it: Euclid's proof of the infinitude of primes. Euclid argued that there must be infinitely many primes as follows: Assume not, and thus let p_1, ..., p_n be all the primes. Consider the product p_1 * ... * p_n + 1; either that number is prime, or it is divisible by a prime p. This prime p cannot be any of p_1, ..., p_n, as each p_i leaves remainder 1. Thus there are infinitely many primes, and we denote this new prime p by p_{n+1}. Lather, rinse, repeat. Keep doing this and we'll get an infinite list of primes. OK, great. This shows there are infinitely many. What can we say about the sequence of primes constructed? Does it contain all the primes? Do we know which primes are in the list and when? Is it easy to compute the terms? Euclid's method leads to the following sequence of primes: 2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, 2801, 11, 17, 5471, 52662739, 23003, 30693651606209, 37, 1741, 1313797957, 887, 71, 7127, 109, 23, 97, 159227, 643679794963466223081509857, 103, 1079990819, 9539, 3143065813, 29, 3847, 89, 19, 577, 223, 139703, 457, 9649, 61, 4357....   (Remember how we generated the sequence. We started with p_1 = 2, the first prime. We apply Euclid's argument and consider 2+1; this is the prime 3 so we set p_2 = 3. We apply Euclid's argument and now have 2*3+1 = 7, which is prime, and set p_3 = 7. We apply Euclid's argument again and have 2*3*7+1 = 43, which is prime and set p_4 = 43. Now things get interesting: we apply Euclid's argument and obtain 2*3*7*43 + 1 = 1807 = 13*139, and set p_5 = 13.) This is a great sequence to think about, but a computational nightmare to enumerate! I downloaded these terms from the Online Encyclopedia of Integer Sequences (homepage is http://oeis.org/  and the page for our sequence is http://oeis.org/A000945). You can enter the first few terms of an integer sequence, and it will list whatever sequences it knows that start this way, provide history, generating functions, connections to parts of mathematics, .... This is a GREAT website to know if you want to continue in mathematics. There have been several times I've computed the first few terms of a problem, looked up what the future terms could be (and thus had a formula to start the induction).
  • Some other fast algorithms (much faster than brute force).
    • Horner's algorithm to evaluate a polynomial quickly: 4x^3 + 5x^2 - 3x + 8 = ((4x+5)x - 3)x + 8 (saves a few multiplications!). Saving multiplications is very important; one application of evaluating polynomials quickly is in constructing Mandelbrot sets.
    • Fast matrix multiplication: Naively we expect it takes \(N^3\) multiplications to find all \(N^2\) entries of \(A^2\) or \(AB\) when \(A\) and \(B\) are \(N\times N\) matrices. The Strassen algorithm (see also the Mathworld entry here, which I think is a bit more readable) does it in about \(N^{\log_2 7}\); the reason for this savings is that they can multiply two \(2\times 2\) matrices with seven and not 8 multiplications (\(3 = \log_2 8\)). The best known algorithm is the Coopersmith-Winograd algorithm, which is of the order of \(N^2.376\) multiplications. See also this paper for some comparison analysis, or email me if you want to see some of these papers.
      • Some important facts. The Strassen algorithm has some issues with numerical stability.
      • One can ask similar questions about one dimension matrices, i.e., how many bit operations does it take to multiply two \(N\) digit numbers. It can be done in less than \(N^2\) bit operations (again, very surprising!). One way to do this is with the Karatsuba algorithm (see also the Mathworld entry for the Karatsuba algorithm).
  • We mentioned the Fundamental Theorem of Algebra (my favorite proof uses complex analysis, which I'll teach next year in that course). We saw it fails miserably in \(Z/nZ\). While there are closed form solutions for quadratics and cubics and quartics in terms of the coefficients, there is no such formula for a general degree 5 or higher polynomial. Interestingly if we add \(i = \sqrt{-1}\) to the reals we get the complex numbers; it's fascinating that adding just one number (and linear combinations) allows us to solve all polynomials with complex coefficients; we don't need to add something new to solve \(x^3 = 1\). Show in a similar spirit that once we can prove there is no formula for the roots of one qunitic in terms of the coefficients of the polynomials that for any degree \(d \ge 6\) there is at least one polynomial where we cannot write the roots in terms of the coefficients. If you do this email me for extra credit.
  • Finally, we talked about the Chinese Remainder Theorem. The statement was so detailed it actually gave us the proof!

  Wednesday, February 15. Video: Clock Arithmetic, Binomial Coefficients, Morley's theorem, Fermat's little Theorem: https://youtu.be/46m_kBnrzeo

  • We reviewed clock and modular arithmetic: https://en.wikipedia.org/wiki/Modular_arithmetic
  • We discussed the notion of a group and gave examples: https://en.wikipedia.org/wiki/Group_(mathematics) and gave as a fun example the Rubik's cube and mentioned the G-d number for the cube (any cube can be solved in at most 20 moves).
  • We talked about Morley's theorem as an example of how brute force proofs can be unenlightening. If you want to see my proof let me know.
  • We spent a lot of time on Fermat's little Theorem, giving two proofs.
    • Our proofs used the factorial function, and one also used binomial coefficients.
    • The factorial function can be generalized to the Gamma function. There's another generalization from a former officemate of mine, Manjul Bhargava.
    • We used the binomial theorem; you might remember this as a key ingredient in calculus....

• Monday, February 13. Video: Divisibility tests, clock arithmetic, unique factorization of primes, Riemann zeta function: https://youtu.be/Qmz9BAk8SrM

  • Sieve of Eratosthenes
  • Paper by Miller and Silva: When a prime divides a product
  • Divisibility rules: https://en.wikipedia.org/wiki/Divisibility_rule
  • Clock or modular arithmetic: https://en.wikipedia.org/wiki/Modular_arithmetic

  • We then moved to the Riemann zeta function, zeta(s), one of the most important functions in all of mathematics. It falls into the category of a generating function, and allows us to pass from local information to a global object, from which we can extract a lot of information. 

  • There's far too much to say about zeta(s) for a class, but here are a few highlights.
    • The fact that \(sum_{n = 1}^\infty 1/n^2 = \pi^2/6\) has a lot of applications. It can be used to prove that there are infinitely many primes via the Riemann zeta function. The Riemann zeta function is \(\zeta(s) = \sum_{n = 1}^\infty 1/n^s\). By unique factorization (also known as the Fundamental Theorem of Arithmetic), it also equals \(\prod_{p\ {\rm prime}} 1 / (1 - 1/p^s)\); notice that a generalization of the harmonic sum and the geometric series formula are coming into play. It turns out that \(\zeta(2) =  \pi^2/6\), as can be seen in many different ways. As \(\pi^2\) is irrational, if there were only finitely many primes then the product would be irrational, contradiction! See wikipedia for a proof of this sum.
      • 6/pi^2 is the probability that two randomly chosen integers are relatively prime.
      • It's also the probability a randomly chosen integer is square-free!
    • We can also use the fact that \(\sum_{n=1}^\infty 1/n\) diverges to prove there are infinitely many primes: if there were only finitely many primes then as \(s \to 1\) from above the product would be finite, and that contradicts the sum diverging! We took logs of the truncated zeta function and used the fact that \(\sum_{n < x} 1/n \approx \log(x)\) (the harmonic numbers) to show that \(\sum_{p < x} 1/p \approx \log\log(x)\). What this shows is that it's easier to count the primes with a \(1/p\) weight than weighting all equally; we then pass to what we desire by partial summation (the discrete version of integration by parts). It turns out significantly better weights are \(\log(p)\) (over \(1/p\)); to see this requires more complex analysis (it essentially arises by taking the logarithmic derivative, a very natural process in complex analysis).
      • A large amount of modern mathematics involves taking a series that only converges for certain values and `extending' it. For example, \(1 + 2 + 4 + 8 + 16 + \cdots\) really does, in some sense, equal -1. This is the principle of meromorphic continuation, one of the central themes of complex analysis. We can similarly continue the zeta function, and we find \(\zeta(-1) = 1 + 2 + 3 + 4 + \cdots = -1/12\).
    • Another interesting application of summing series involving primes is to the Pentium bug (see the links there for more information, as well as Nicely's webpage). The calculation being performed was \(\sum\) \((p: p\ \) prime and either\ \(p+2\) or \(p-2\) is prime}} \(1/p\); this is known as Brun's constant. If this sum were infinite then there would be infinitely many twin primes, proving one of the most famous conjectures in mathematics; sadly the sum is finite and thus there may or may not be infinitely many twin primes (twin primes are two primes differing by 2).

• Friday, February 10. Video: Axioms, Division and Euclidean Algorithms: https://youtu.be/BZwacoCYU4Y

  • Well ordering principle   and   Russell's paradox   and    Axiom of Choice   and  Continuum hypothesis

    • Russell's paradox is one of the most famous in all of mathematics; it showed that we didn't even understand what it meant to be a set or an element of a set! Another famous one is the Banach - Tarski paradox, which tells us that we don't understand volumes! It basically says if you assume the Axiom of Choice, you can cut solid sphere into 5 pieces, and reassemble the five pieces to get two completely solid spheres of the same size as the original! While it is rare to find these paradoxes in mathematics, understanding them is essential. It is in these counter-examples that we find out what is really going on. It is these examples that truly illuminate how the world is (or at least what our axioms, imply). Most people use the Zermelo-Fraenkel  axioms, abbreviated ZF. If you additionally assume the Axiom of Choice, it's called ZFC or ZF+C. Not all problems in mathematics can be answered yea or nay within this structure. For example, we can quantify sizes of infinity; the natural numbers are much smaller than the reals; is there any set of size strictly between? This is called the Continuum Hypothesis, and my mathematical grandfather (my thesis advisor's advisor), Paul Cohen, proved it is independent (ie, you may either add it to your axiom system or not; if your axioms were consistent before, they are still consistent).
    • In a real analysis course, one develops the notation and machinery to put calculus on a rigorous footing. In fact, several prominent people criticized the foundations of calculus, such as Lord Berkeley; his famous attack, The Analyst, is available here. It wasn't until decades later that a good notion of limit, integral and derivative were developed. Most people are content to stop here; however, see also Abraham Robinson's work in Non-standard Analysis. He is one of several mathematicians we'll encounter this semester who have been affiliated with my Alma Mater, Yale. Another is the great Josiah Willard Gibbs.
  • Division algorithm
  • Euclidean algorithm
    • There are many processes in mathematics that run in far less time than they appear. Below is a quick description of a few, many of which we've already seen (but this is a good time to revisit). The Euclidean algorithm is a terrific example. It runs far faster than expected; it doesn't take \(\min(x,y)\) steps but rather \(2 \log(\min(x,y))\). It's a beautiful example of a major theme of the course, the need to do things fast. The other topic was linearization non-linear terms. This was a major theme of calculus (Newton's method is a terrific example). Here are some other examples.
      • Horner's algorithm to evaluate a polynomial quickly: \(4x^3 + 5x^2 - 3x + 8 = ((4x+5)x - 3)x + 8\) (saves a few multiplications!). Saving multiplications is very important; one application of evaluating polynomials quickly is in constructing Mandelbrot sets (see below when we discuss Newton's method).
      • Telescoping sums: often re-arranging the algebra leads to a significantly easier computation.
      • Fast matrix multiplication: Naively we expect it takes \(N^3\) multiplications to find all \(N^2\) entries of \(A^2\) or \(AB\) when \(A\) and \(B\) are \(N\times N\) matrices. The Strassen algorithm (see also the Mathworld entry here, which I think is a bit more readable) does it in about \(N^{\log_2 7}\); the reason for this savings is that they can multiply two \(2\times 2\) matrices with \(7\) and not \(8\) multiplications (\(3 = \log_2 8\)). The best known algorithm is the Coopersmith-Winograd algorithm, which is of the order of \(N^{2.376}\) multiplications. See also this paper for some comparison analysis, or email me if you want to see some of these papers.
        • Some important facts. The Strassen algorithm has some issues with numerical stability.
        • One can ask similar questions about one dimension matrices, ie, how many bit operations does it take to multiply two \(N\) digit numbers. It can be done in less than \(N^2\) bit operations (again, very surprising!). One way to do this is with the Karatsuba algorithm (see also the Mathworld entry for the Karatsuba algorithm).
      •  Newton's method is significantly more powerful than divide and conquer (also called the bisecting algorithm); this is not surprising as it assumes more information about the function of interest (namely, differentiability). The numerical stability of Newton's method leads to many fascinating problems. One terrific example is looking at roots in the complex plane of a polynomial. We assign each root a different color (other than purple), and then given any point in the complex plane, we apply Newton's method to that point repeatedly until one of two things happen: it converges to a root or it diverges. If the iterates of our point converges to a root, we color our point the same color as that root, else we color it purple. This leads to Newton fractals, where two points extremely close to each other can be colored differently, with remarkable behavior as you zoom in. If you're interested in more information, let me know; a good chaos program is xaos (I have other links to such programs for those interested). One final aside: it is often important to evaluate these polynomials rapidly; naive substitution is often too slow, and Horner's algorithm is frequently used. What we're really doing in Newton's method is replacing a complicated function with a linearization.
        • One of the most important applications of the Euclidean algorithm is in RSA (cryptography) to compute modular inverses, and to determine some of the items needed for our keys.
        • There were several great questions from the class about the efficiency of the Euclidean algorith. We initially looked at it and said it must run in at most \(x\) steps (where \(x < y\) are our two numbers). After some analysis we showed it runs in at most \(2 \log_2(x)\) steps; however, maybe it's even faster, or faster on average. Click here for more on it's runtimes, and the various measures (average run time, worst case scenario). This leads to the topic of computational complexity.
  • Paper by Miller and Silva: When a prime divides a product
  • Ramsey Numbers: https://en.wikipedia.org/wiki/Ramsey's_theorem
    • Erdos quotes: The one from class is: Suppose aliens invade the earth and threaten to obliterate it in a year's time unless human beings can find the Ramsey number for red five and blue five. We could marshal the world's best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded the Ramsey number for red six and blue six, however, we would have no choice but to launch a preemptive attack.  --As quoted in "Ramsey Theory" by Ronald L. Graham and Joel H. Spencer, in Scientific American (July 1990), p. 112-117

• Wednesday, February 8. Video: Basic coding in Mathematica:

  • Purpose of today was to do basic coding. Below are lectures from various classes
    • 2017 Problem Solving (331) / Number Theory (313): Mathematica code   (the pdf)      Video: 313: https://youtu.be/khMJuFXOhFE   Video 331: https://youtu.be/vPGjly_7QVw
    • 2016 Probability Iteration: Click here for the Mathematica code                Click here for a pdf        Video: https://youtu.be/JMhkq_bczKk
    • 2015 Probability Iteration: Click here for the Mathematica code                Click here for a pdf        Video:  https://youtu.be/wUC-iUsZL6M
    • some suggested problems you should try (click here for Mathematica notebook, and click here for a pdf).
  • We discussed the cookie problem, counting the number of ways to divide 10 identical cookies among 5 distinct people (here's the classic clip where Cookie Monster meets the Count, whose full name is Count von Count -- they changed how he appears!). Counting is very important; we discussed how we could (and SHOULD) do simpler variants to build intuition. It's usually called the stars and bars problem. What I love here is the power of changing your perspective -- we go from a very painful brute force approach to being able to solve it in one line.
    • I have posted the cookie problem on my math riddles page (email me if you want to contribute); someone with far more patience than I solved it by brute force. Here's their solution. The final number, tabbed from the others, are how many distinct rearrangements we have of this basic configuration. The total  is 1001, or (10+5-1 choose 5-1). We saw this problem lead to a discussion of multinomial coefficients. When there are two people getting cookies, say 8 and 2, there aren't (5 choose 2) ways to assign people, but (5 choose 2) * 2! (we choose the 2 people, then there are 2! ways to choose which gets the 8 and which gets the 2). If we have 8 1 1 it's more involved. In that case it's (5 choose 3) to choose the three people, 3! ways to order which of the people gets which number, but then we must divide by 2! (as the two people getting 1 are indistinguishable). A better way to view 3!/2! is 3! / (2! 1!) (note the numbers on the bottom sum to the top). This is an example of a multinomial coefficient, a generalization of binomial coefficients. For example, if we have MISSISSIPPI, there would be 11! ways to order the letters (order matters) if the letters are distinguishable, but they're not. So let's put subscripts on the letters: MI₁S₁S₂I₂S₃S₄I₃P₁P₂I₄. We then have 4! ways of placing the four marked S's in the four S positions, and so on, giving 11! / (4! 4! 2! 1!) (I like including the final 1 so that the bottom sums to the top).
    • Below are the person's solutions, arranged differently than we did in class.
    • 0 0 0 0 10         5
      
      0 0 0 1 9         20
      0 0 0 2 8         20
      0 0 0 3 7         20
      0 0 0 4 6         20
      0 0 0 5 5         10
      
      0 0 1 1 8         30
      0 0 1 2 7         60
      0 0 1 3 6         60
      0 0 1 4 5         60
      0 0 2 2 6         30
      0 0 2 3 5         60
      0 0 2 4 4         30
      0 0 3 3 4         30
      
      0 1 1 1 7         20
      0 1 1 2 6         60
      0 1 1 3 5         60
      0 1 1 4 4         30
      0 1 2 2 5         60
      0 1 2 3 4         120
      0 1 3 3 3         20
      0 2 2 2 4         20
      0 2 2 3 3         30
      
      1 1 1 1 6         5   
      1 1 1 2 5         20
      1 1 1 3 4         20
      1 1 2 2 4         30
      1 1 2 3 3         30
      1 2 2 2 3         20
      
      2 2 2 2 2          1
    • What we're really doing is solving the equation \(x_1 + \cdots + x_5 = 10\) in non-negative integers. This is a very special type of Diophantine equation. It's actually a special case of Waring's problem, which looks at solving \(x_1^k + \cdots + x_s^k = n\) for fixed \(s\) and \(k\). These problems are in general not accessible through combinatorics; the case \(k=1\) is special. The general approach proceeds via generating functions, which we will cover in great detail later in the semester (it's one of the key concepts of the class).
    • Our solution to the cookie problem is quite elegant, and in some respects reminiscent of geometry class (remember all those proofs where the teacher cleverly adds auxiliary lines; the difference here is we just add more cookies). While it is possible to solve many combinatorial problems by brute force in principle, in practice this is not a good way to go -- it is time consuming, and quite likely that one makes a mistake. Typically one finds a way to interpret a given quantity two ways; we can compute one of them and thus we obtain a formula for the other. For example, we showed the number of ways of dividing C cookies among P people is (C + P - 1 choose P-1); here all the identical cookies are divided. What if we don't assume all the cookies are divided -- what is the answer now? It is just Sum_{c = 0 to C} (c + P - 1 choose P - 1); this is because we are just going through all the cases (we give out no cookies, 1 cookie, ...). What does this sum equal? Imagine now we have another person, say the Cookie Monster (this is one of Cameron's favorite clips), who gets all the remaining cookies. Then dividing at most C cookies among P people is the same as dividing exactly C cookies among P+1 people, and hence our sum equals (C + P+1 - 1 choose P+1 - 1).
    • Related to the cookie problem is the partition problem: for the cookie problem we consider 2+3+3+1+1 different from 1+2+3+3+1 as the five people are distinct; if we don't consider these distinct then we have a partition problem. It's a lot more complicated to count these, but some great mathematics (such as Young tableau).

• Monday, February 6. Video: Algorithms, notation, Babylonian math, Gregory-Leibniz formula: https://youtu.be/4ku0APonNxc

  • Factorization: https://en.wikipedia.org/wiki/Factorization
  • Sudoku: https://en.wikipedia.org/wiki/Sudoku
  • Strassen algorithm for fast multiplication of matrices: https://en.wikipedia.org/wiki/Strassen_algorithm
  • We justified studying primes by talking about how they could be used in cryptography, especially the power to create an encryption system where the sentry will only let in the right people but cannot be tortured as it does not know the password! Later in the semester we'll find a fast way to often tell if a number is composite without finding any factors!
  • We talked about how the Babylonians did mathematics. In addition to the horrors of base 60 (which Wikipedia tells me is due to the Sumerians, which must be true if they've posted it), they did give us the look-up table. The point is to reduce long, painful calculations to pre-computed quantities, with perhaps some (hopefully linear) interpolation as you can't pre-compute everything. In base 60, one would need to have tables for about 3600/2 = 1800 multiplications to be able to do \(xy\); the Babylonians noticed \(xy = ((x+y)^2 - x^2 - y^2) / 2\), or even better \(xy = ((x+y)^2 - (x-y)^2)/4\); this reduces the problem to knowing just squares (only 60 entries needed) and the ability to subtract and divide by 2. It's much better to do these simpler problems than the original harder one.
  • Diophantine equaitons: https://en.wikipedia.org/wiki/Diophantine_equation
  • Gregory-Leibniz formula for \(pi\): https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80  (we were able to get here through the chain rule applied to inverse functions, the geometric series formula, ...; note that work is required to justify some of our steps). The function \(1/(1+x^2)\), for \(-\infty < x < \infty\), is very useful in probability. If we multiply by \(1/\pi\) it integrates to 1 and becomes the Cauchy distribution. What makes this so nice is that while it is arguable if has a mean, it clearly has infinite variance and thus provides a great testing ground for some conjectures (i.e., what conditions are truly needed).
  • Geometric series: https://en.wikipedia.org/wiki/Geometric_series
    • An infinite series of surprises: a nice article going from the geometric series to the harmonic series to other important examples.
    • The geometric series formula only makes sense when \(|r| < 1\), in which case \(1 + r + r^2 + \cdots = 1/(1-r)\); however, the right hand side makes sense for all r other than 1. We say the function \(1/(1-r)\) is a(meromorphic) continuation of \(1+r+r^2+\cdots.\) This means that they are equal when both are defined; however, \(1/(1-r)\) makes sense for additional values of \(r\). Interpreting \(1+2+4+8+\cdots\) as \(-1\) or \(1+2+3+4+5+ \cdots = -1/12\) actually DOES make sense, and arises in modern physics and number theory (the latter is \(\zeta(1)\), where \(zeta(s)\) is the Riemann zeta function)!
    • There is a very nice proof using memoryless processes.... 

     

• Friday, February 3. First lecture: slides here       handout here       video here: https://youtu.be/-swSmjLqlfE

  • To highlight the difference between theoretically useful and computationally useful, we looked at Euclid's proof of the infinitude of primes. If there are only finitely many primes, say \(p_1, \dots, p_n\), then consider \(p_1 \cdot p_2 \cdots p_n + 1\); either this is a new prime or it is divisible by a prime not in our list, since each prime in our list has remainder 1 when we divide by it.
  • Euclid's argument actually gives a lower bound, namely on the order of \(\log \log x\) primes at most \(x\) (the true answer is that there are about \(x / \log x\) primes at most \(x\)). As a nice exercise (for fun), prove that Euclid's argument gives a bound of this size.
  • This leads to an interesting sequence: 2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, 2801, 11, 17, 5471, 52662739, 23003, 30693651606209, 37, 1741, 1313797957, 887, 71, 7127, 109, 23, 97, 159227, 643679794963466223081509857, 103, 1079990819, 9539, 3143065813, 29, 3847, 89, 19, 577, 223, 139703, 457, 9649, 61, 4357....   This sequence is generated as follows. Let a_1 = 2, the first prime. We apply Euclid's argument and consider 2+1; this is the prime 3 so we set \(a_2 = 3\). We apply Euclid's argument and now have \(2\cdot 3+1 = 7\), which is prime, and set \(a_3 = 7\). We apply Euclid's argument again and have \(2\cdot 3\cdot 7+1 = 43\), which is prime and set \(a_4 = 43\). Now things get interesting: we apply Euclid's argument and obtain \(2\cdot 3\cdot 7 \cdot 43 + 1 = 1807 = 13\cdot 139\), and set \(a_5 = 13\). Thus \(a_n\) is the smallest prime not on our list genereated by Euclid's argument at the nth stage. There are a plethora of (I believe) unknown questions about this sequence, the biggest of course being whether or not it contains every prime. This is a great sequence to think about, but it is a computational nightmare to enumerate! I downloaded these terms from the Online Encyclopedia of Integer Sequences (homepage is http://oeis.org/ and the page for our sequence is http://oeis.org/A000945 ). You can enter the first few terms of an integer sequence, and it will list whatever sequences it knows that start this way, provide history, generating functions, connections to parts of mathematics, .... This is a GREAT website to know if you want to continue in mathematics. There have been several times I've computed the first few terms of a problem, looked up what the future terms could be (and thus had a formula to start the induction).
  • We're probably the only college in the world to talk about this in a class on operations research (I did so last semester), but I think it's a great use of time for many reasons, ranging from seeing how fast algorithms run, to how well they run (is anything missed), to the difficulties of finding them (hey, this class is all about finding solutions -- how do we find the terms in this sequence?). For more on these primes, see here:
    • http://en.wikipedia.org/wiki/Euclid%E2%80%93Mullin_sequence
    • http://www.jstor.org/stable/2044665?origin=crossref
    • http://arxiv.org/pdf/1107.3318v2.pdf (this shows a variant omits infinitely many primes, proved by one of my mathematical brothers)
    • http://www.math.uga.edu/~pollack/mullin.pdf (elementary proof of the above)
  • There are other proofs of the infinitude of primes. In a real analysis course, one develops the notation and machinery to put calculus on a rigorous footing. In fact, several prominent people criticized the foundations of calculus, such as Lord Berkeley; his famous attack, The Analyst, is available here. It wasn't until decades later that a good notion of limit, integral and derivative were developed. Most people are content to stop here; however, see also Abraham Robinson's work in Non-standard Analysis. One of my favorite applications of open and closed sets is Furstenberg's proof of the infinitude of primes; one night while a postdoc at Ohio State I had drinks with Hillel Furstenberg and one of his students, Vitaly Bergelson. This is considered by many to be one of the best proofs of the infinitude of primes; it is so good it is one of six proofs given in THE Book. Unlike most proofs of the infinitude of primes, this gives no bounds on how many primes there are at most x.
  • Whenever you are given a new theorem (such as the Fundamental Theorem of Calculus), you should always check its predictions against some cases that you an readily calculate without using the new machinery. For example, if we want to find the area under \(f(x)\) from \(x=0\) to \(x=1\), obviously the answer will depend on \(f\). If \(f\) is constant it is trivial (area of a rectangle).
    • If \(f\) is a linear relation then the answer is still readily calculated (area of a triangle). We can check by using our result on sums of integers: \(\sum_{k=1}^n k = n(n+1)/2\). The simplest way to prove this is to call the sum \(S_n\), write it backwards, and notice that we now have \(k\) and \(n+1-k\) paired; thus we have \(n\) sums of value \(n+1\), so \(2 S_n = n(n+1)\) or \(S_n = n(n+1)/2\).
    • For more general polynomials, one can compute the Riemann sums (the upper and lower sums) by Mathematical Induction. For example, using induction one can show that \(\sum_{k=1}^n k^2 = n(n+1)(2n+1)/6\), and this result can then be used to find the area under the parabola \(y = x^2\). What do you think \(\sum_{k=1}^n k^3\) will be? It can be shown that \(\sum_{k=1}^n k^m\) is a polynomial of degree \(m+1\) in \(n\), with leading coefficient \(\frac{1}{m+1}\). This is quite reasonable, as for \(n\) large if we look at \(\sum_{k=1}^n (k/n)^m \cdot (1/n)\) this looks like \(\int_0^1 x^m dx\), and the anti-derivative of \(x^m\) is \(x^{m+1}/(m+1)\). If you're interested in how you prove that this is a polynomial, let me know. Here's one intriguing hint: if you assume it is a polynomial in \(n\) then if you evaluate it for enough values of \(n\) you can interpolate and figure out the polynomial, and then feed that into the mathematical induction!