april fools

perhaps this is just illustrates the inner geek in me more fully, but the following things have greatly amused me today:

   
• NASA’s photo of the day has an awesome caption.
     
• the Google empire has gone crazy... gmail offers a "send emails from the past" option, google calendar is offering a "wake up kit" and an "i’m feeling lucky" option to set up appointments with celebrities, youtube is rickrolling all the featured videos... and apparently they’re starting a colony on mars. hmmm... and google australia is letting you search the future with gDay.  the testimonials are funny.
     
• my advisor, already famous for his online opinions, continues his almost-annual april fool’s opinion tradition.
     
• and finally..., speaking of my advisor, yesterday he sent out the following email:
  
  
  Dear Math 640 students,
  
  I just finished posting two homework sets. The first one is due tomorrow (please slide them under my door), and the second one is due Thurs. I am rather disappointed at some of you who have been late handing-in the homework (and I commend those who always hand them in on time).
  
  Anyway, the homework problem set that is due tomorrow should be completed by tomorrow, 11:59pm, or else you would get zero, and be in danger of failing this class (i.e. getting a B).
  
  
  this is already suspicious, since we know he loves april fools pranks, but it was worth figuring out anyhow... the problem set he posted for today is:
  ~~~~~~~~~~~~
  

  First Homework Set for March 31, 2008 class, Due April 1, 2008
  [No extensions!]
  

  
  
  
  
  • Recall that for any set of non-negative integers A, mex(A) is
    the smallest  non-negative integer not in A. For example,
    mex({2,4,5})=0, mex({0,1,2,5,8})=3, etc.
    
    Define a sequence a_i recursively by a₁=2, and for i >= 1  by:
    a_i=mex({0,1} U { j a_r, j >= 1 , 1 <= r < i}),
    
    Prove the following properties of a_i
    
    
    
    1. There are infinitely many i such that a_i+1-a_i=2
       
    2. Every even intger n >= 6 can be written as
       a_i+a_j, for some pos. integers i and j.
       
    3. Define a sequence F(n) by,
       F(a_i1 a_i1 a_i2  ...a_ir)=(-1)^r if n can be expressed as a product of distinct a_i’s , and 0 otherwise.
       Let G(n)=add(F(i), i=0..n)
       Prove that |G(n)| <= Cn^.999, for some fixed constant C.
       
    
    
  • Remember that Euler’s pentagonal product
    eta(q)=(1-q)(1-q²)(1-q³) ... ,
    when expanded, has lots of 0-coefficients and the rest are 1 or -1.
    Condider the 24-th power of that
    eta(q)²⁴=[(1-q)(1-q²)(1-q³) ...]²⁴,
    and let’s call the coeff. of qⁿ,  tau(n). Prove that tau(n) is never zero.
    
  
  ~~~~~~~~~~~~~~~
  thus, almost instantly, emails started flying between his ph.d. students as follows:
  ~~~~~~~~~~~
  
  is it an april fools joke? the homework that he posted looks hard. i don’t want to do it.
  -em
  
  ~~~~~~~~~~~
  
  i assume it is.  i was just wondering if it was equivalent to one of the clay problems or something equally ridiculous ;) why else would it be due on april 1?
  lara
  
  ~~~~~~~~~~~
  and finally...
  
  Don’t read below if you haven’t looked at the homework.  To make a buffer, here’s the Millenium Falcon:
  
                   c==o
  
                 _/____.._
  
          _.,--’" ||^ || "..z._
  
         /_/^ ___..||  || _/o.. "..-._
  
       _/  ]. L_| || .||  .._/_  . _..--._
  
      /_~7  _ . " ||. || /] .. ]. (_)  . "..--.
  
     |__7~.(_)_ []|+--+|/____T_____________L|
  
     |__|  _^(_) /^   __..____ _   _|
  
     |__| (_){_) J ]K{__ L___ _   _]
  
     |__| . _(_) ..v     /__________|________
  
     l__l_ (_). []|+-+-<..^   L  . _   - ---L|
  
      ..__..    __. ||^l  ..Y] /_]  (_) .  _,--’
  
        ..~_]  L_| || ... ...../~.    _,--’"
  
         .._.. . __/||  |..  ....-+-<’"
  
           "..---._|J__L|X o~~|[....     
  
              ..____/ ..___|[//     
  
                   ..--’   ..--+-’
  "Millenium Falcon" Modified Corellian YT-1300 Transport
  
  (No, I did not draw that - I don’t have the patience for ASCII art).
  
  It’s certainly April Fool’s Joke.  I didn’t recognize that a(n) is the nth prime at first, but now I see that his problems are:
        1. The Twin Prime Conjecture
        2. Goldbach’s conjecture
        3. His G is the Merten’s function (I had to look this up).  This result
        would be equivalent to the Riemann Hypothesis, apparently.
  
  The partition-related function is Ramanujan’s tau function.  That tau(n) is never zero is another unsolved conjecture.  It’s a shame - I was kind of hoping this one would be a ridiculously easy problem, but that everyone would be scared off by the first three.
  
  -Baxter
  
  

hooray for clever fun. ;)