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Probabilistic Number Theory Project Description Prerequisites Warm up Problems Hints Resources Teaching Notes Extension Problems Results

Teaching Notes for the
Probabilistic Number Theory Problem

If you want to carry it to completion, this project will require a fair amount of teaching. Think of it in five phases:

  1. Introduction: what are the problems asking?
  2. Perfect squares: a complete solution.
  3. Empirical investigations: running simulations.
  4. Mathematizing: translating to infinite products and sums.
  5. Analysis: evaluating the products and sums.

Students can get a good research experience by carrying out the first three items; the last two are rather advanced and will stretch your best students. In addition, some students will want to take a detour to look at things like the fundamental theorem of arithmetic--the fact that every integer can be factored in essentially one way into a product of primes. This is discussed on The Fundamental Theorem of Arithmetic.

So, there are many ways to do this. Here’s just one possibility.

Phase 1.

What does probability look like in infinite sample spaces? Discuss the difference between “probability 0” and “impossible” (see Results). Work through some “dartboard” examples of geometric probability. This could take two or three classes. Then work through the Warm Up Problems. That will take another class. It’s really important at this stage to get students talking and writing about their thinking so you can catch the subtle (and inevitable) misconceptions that develop. For example, get them to explain their intuition about the probability of getting “3” on a random pick of integers. Why is this different from “getting 3 if the pick is restricted to the integers between 1 and 100?”

Expect unusual and clever ideas. One (very advanced) student argued that the probability of picking the integer n should be 12n-. She had two reasons why this was a good choice:

  • The sum of the probabilities over all the integers is 1.
  • If someone asks you to pick a number at random, you’re more likely to pick a small number than a very large one, so probability should decrease with size.

Her model led to some very strange consequences.

Phase 2.

Work through the perfect square probability. Ask students to come up with a formula for the number of perfect squares between 1 and n (see Results). After students have ideas that are general in principle, you may have to introduce the greatest integer function and talk a bit about limits. Estimate three classes for this.

Phase 3.

Build and run a simulation. This could take anywhere from a couple days to a couple weeks, depending on how facile your students are with the relevant technology. Try to build the simulation so that the “test” is modular--that is, the same simulation should estimate the probability that an integer is prime or square, just by changing the tester. This is an especially nice project for a computer science class.

If you are using Mathematica, a crude start on a program that lists the primes in a given range goes something like this:
 g[n_] := If[PrimeQ[n] == True,1,0]*n
This models a function g defined by

      {
g(n) =  n    if n is prime
        0    if n is not prime
Then define f that maps g over the integers from 1 to m:
 f[m_] := Map[ g,Range[m]]
So, for example, f(100) outputs a list of 100 numbers that contains all the primes between 1 and 100 (and some zeros).

Phase 4.

Here’s where more direct teaching begins. Building on the Warm Up Problems, you’d like students to reason that the probability that a number between 1 and 100 is prime is
(    1)(    1 )(    1)(     1)
 1-  2   1- 3    1- 5   1 - 7
This is because a composite number between 1 and 100 has a prime factor between 1 and  V~ --
 100. So, a number between 1 and 100 is prime if and only of it is not divisible by 2, 3, 5, or 7. And you’d like things to generalize from here.

You can expect things like this:

The number of integers not divisible by 2 is half the total. The number not divisible by three is 23 of the total. The number not divisible by four is 34 of the total. The number not divisible by five is 45 of the total. Primes aren’t divisible by any numbers (apart from one and themselves of course). Maybe there’d be a way to work out the proportion of primes through the above? I’m going to have a longer think about that...

A brilliant (but incorrect) idea. But it’s the germ of exactly the right idea, and you’ll have to find a way to tease out the subtle mistake (the 34). What’s worse is that this brilliant mistake is more than likely not the one that will happen with your students. Think about several classes (maybe a week) to get this solid, and beware that, even then, it won’t be solid for some students.

The point here is to get students comfortable with writing down a mathematical expression that, if evaluated, would give the right probability. So, this phase is as much about building confidence in the technique as it is about developing skill at the technique. For example, the problem above (pick a prime less than 100) can be “checked” in a couple ways: Have one group of students evaluate the product and another count the primes between 1 and 100. Then try the same experiment (using a CAS) with a sample space of 1-500. When you get to the infinite product, you have a real laboratory for experimentation.

You might ask students to compare

the-number-of primes-up-to n-
            n
with
           (     )
     prod           1
         V~ - 1 - p
primes up to n
for some values of n between 1 and 1000. This “product notation” is similar to summation notation. For example,
      prod       (     )        prod      (     )   (     )(      )(      )(      )
              1 - 1  =             1-  1  =  1 - 1   1 - 1   1-  1   1- 1   = -8  ~~  .22857
primes up to V~ 100  p    primes up to 10  p         2       3       5      7     35
Here’s a table of the first 168 primes (the primes up to 1000). It was generated in Mathematica by the command

TableForm[Table[ {n,Prime[n]},{n,1,168}]]

 

n the nth prime


1 2


2 3


3 5


4 7


5 11


6 13


7 17


8 19


9 23


10 29


11 31


12 37


13 41


14 43


15 47


16 53


17 59


18 61


19 67


20 71


21 73


22 79


23 83


24 89


25 97


26 101


27 103


28 107


29 109


30 113


31 127


32 131


33 137


34 139


35 149


36 151


37 157


38 163


39 167


40 173


41 179


42 181


n the nth prime


43 191


44 193


45 197


46 199


47 211


48 223


49 227


50 229


51 233


52 239


53 241


54 251


55 257


56 263


57 269


58 271


59 277


60 281


61 283


62 293


63 307


64 311


65 313


66 317


67 331


68 337


69 347


70 349


71 353


72 359


73 367


74 373


75 379


76 383


77 389


78 397


79 401


80 409


81 419


82 421


83 431


84 433


n the nth prime


85 439


86 443


87 449


88 457


89 461


90 463


91 467


92 479


93 487


94 491


95 499


96 503


97 509


98 521


99 523


100 541


101 547


102 557


103 563


104 569


105 571


106 577


107 587


108 593


109 599


110 601


111 607


112 613


113 617


114 619


115 631


116 641


117 643


118 647


119 653


120 659


121 661


122 673


123 677


124 683


125 691


126 701


n the nth prime


127 709


128 719


129 727


130 733


131 739


132 743


133 751


134 757


135 761


136 769


137 773


138 787


139 797


140 809


141 811


142 821


143 823


144 827


145 829


146 839


147 853


148 857


149 859


150 863


151 877


152 881


153 883


154 887


155 907


156 911


157 919


158 929


159 937


160 941


161 947


162 953


163 967


164 971


165 977


166 983


167 991


168 997


Phase 5.

You’re looking at a month or two for this, maybe longer. Study the Results and you’ll see that there’s some very difficult material here. We’ve found it best to move around--introduce something new, then go back and reinforce something from the earlier phases. Paul Goldenberg (a member of the Making Mathematics team) distinguishes between the “clock time” and “calendar time” it takes to understand something. This takes calendar time--slow but steady hammering at the ideas, every day adding a little new territory while and making more permanent structures on what has already been settled.

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