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## Teaching Notes for theProbabilistic 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 . 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

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
This is because a composite number between 1 and 100 has a prime factor between 1 and . 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 of the total. The number not divisible by four is of the total. The number not divisible by five is 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 ). 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

with
for some values of n between 1 and 1000. This “product notation” is similar to summation notation. For example,
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.

 Translations of mathematical formulas for web display were created by tex4ht. © Copyright 2003 Education Development Center, Inc. (EDC)