# The Teacher s Circle Number Theory, Part 1 Joshua Zucker, August 14, 2006

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2 free to use some smaller numbers; alternatively, if division practice and divisibility tests aren t your goal, let them use calculators: (a) Is 221 prime? If not, what are its factors? (b) What s the largest divisor you need to check to be sure that 397 is prime? How do you know it s the largest? (c) Is 8171 prime, or not? How do you know for sure? A student version of the handout, with slightly different directions than the teacher version on the back of this page, can be found on the back of this packet.

3 Sieve of Eratosthenes (Teacher Version) To use this method, start with a long list of numbers like the one on this page. Forget about 1, since it is neither prime nor composite. [The mathematical term for this kind of number is unit]. Now you know 2 is prime, so circle it. Then cross off all the multiples of 2 (4, 6, 8, 10, 12, ) since you know they cannot be prime. Once you finish that, go to the next number after 2 that isn t crossed out [namely 3], and circle it. Cross off its multiples. Repeat this process until you have circled all the primes and crossed out all the composite numbers. There should be convenient patterns to help with a lot of the crossing out. Things to notice while you do this: as you re crossing out, say, all the multiples of 11, which ones are already crossed out? Where s the first one that s not crossed out? What does that tell you about when you can stop crossing out and just circling?

4 Now that you have a list of primes to work with, you can start to notice some patterns. The next few examples contain some exercises to give people some raw material that might serve as motivation or inspiration for the thought that the problems will require. I wrote them a little more with the student vocabulary in mind than with yours; it probably won t take too much work to convert this to a handout you can use with your students. The distribution of primes (what fraction of numbers are prime? How big are the gaps?) 3. You might have noticed that the gaps between primes tend to get bigger as you go along. Fill in the blanks to get one measure of how much bigger they are getting on average. a) Between 1 and there are 10 primes. b) Between 100 and there are 10 primes. c) Between 400 and there are 10 primes. d) The longest run of consecutive composite numbers between 1 and 100 starts with and ends with and is long. e) The longest run of consecutive composite numbers between 401 and 500 starts with and ends with and is long. [By the way, there s a great opportunity to teach about fencepost errors in these last two! If boats leave every 15 minutes, how many leave from 9am to 5pm? 37, not 36! How many numbers are there from 27 through 33? 7, not 6!] 4. Do the gaps between primes get longer and longer, or is there eventually a record-size gap that never gets beaten? Factorials give a great clue about this problem. A factorial, like 7!, means to multiply all the integers from 7 down to 1. So, 7! is , which is the important part, or 5040, which isn t very important at all. a) The number is composite not prime. How can you be sure? Well, if you can find a factor that s not 1 or the number itself, it must be prime. b) Is composite? c) Based on your answers to the previous two problems, there must be at least how many composite numbers in a row, starting from ? d) Name a number which you can be sure is in a string of at least 99 composite numbers in a row. e) Name a number which you can be sure is in a string of at least 999 composite numbers in a row. f) Explain how you can be sure that any record number of composites in a row eventually gets broken.

6 Squares, Pythagoras, and Primes (with some hints of complex numbers!?) 7. A perfect square, or square number, is what you get when you multiply a whole number by itself. a) Just for practice, write down the first 10 perfect squares. b) List all the primes up to 100. For each one, determine whether it can be made by adding up two perfect squares or not. For example, 5 = 4 + 1, but 7 cannot be written as the sum of two perfect squares. For the numbers that can be written as the sum of two perfect squares, keep track of which squares you used. c) Find a pattern that tells you which primes can be written as the sum of two squares. Hint: look at where they land in the big sieve. How do they line up? d) [Some algebra required, and probably you ll want to do the problem on squares from the division section before you do this one.] Explain why all the impossible ones are impossible to write as the sum of two squares. [Hint: look for where the square numbers fall in the big sieve as well.] e) [Warning: really hard!] Explain why all the possible ones really are possible. That is, how do you know the pattern you ve observed keeps working forever? Prime-generating formulas (Mersenne and Euler) 8. Mathematicians have long sought formulas that generate prime numbers, only prime numbers, and ideally every prime number. Here are two formulas that are a few centuries old, that turn out to be quite interesting prime-generating mechanisms even if they don t work quite as the mathematicians originally hoped. a) One famous formula involves the numbers that are one less than the powers of 2. To begin, = 1, a unit, not prime. Then = 3, which is prime = 7, which is prime. But =, which is. b) That formula has a lot better chance of working if the exponent is prime. Try 2 5 1, 2 7 1, and so on. Are all of those primes? How long does the pattern work? [Notice all the great chances to review vocabulary like exponent and power, practice some arithmetic and even long division, and for more advanced students to get into the difference of squares/cubes/nth powers formulas for factoring.] c) Another famous pattern, for which my favorite mathematician Euler gets the credit, is to start with any number, square it, subtract your original number, and then add 41. In other words, n 2 n When n = 1, we get 1 squared is 1, minus the original 1 leaves 0, adding 41 gives 41. When n = 2, we get 2 squared is 4, minus the original 2 leaves 2, plus 41 makes 43. How long can we use this pattern to produce new prime numbers?

7 Sieve of Eratosthenes A sieve is like a strainer. Eratosthenes is a Greek geometer who is famous for being the first to get an accurate estimate of the size of the Earth. To use this method, start with a long list of numbers like the one on this page. Forget about 1, since it is neither prime nor composite. [The mathematical term for this kind of number is unit]. Now you know 2 is prime, so circle it. Then cross off all the multiples of 2 (4, 6, 8, 10, 12, ) since you know they cannot be prime. Once you finish that, go to the next number after 2 that isn t crossed out [namely 3], and circle it. Cross off its multiples. Now circle the next number that s not crossed out, namely 5, and then cross out all of its multiples. Repeat this process until you have circled all the primes and crossed out all the composite numbers. There should be convenient patterns to help with a lot of the crossing out. Now you have a list of primes to use on the rest of the problems in this section! Think about your warm-ups when you do this, and you ll be able to stop worrying about crossing out, and only circle the rest, after you reach a certain point: what is that point, and why?

9 algorithm is.] I d explain it to students something like this. If you want to find the greatest common divisor of 84 and 66, first you divide 84 by 66 and find the remainder of 18. Euclid understood that the GCD of the original two numbers is the same as the GCD of one of the original two numbers and the remainder [how can we understand this?]. Since smaller numbers are easier to deal with, then we can say GCD(84,66) = GCD(66,18). And now we can repeat the process! 66 divided by 18 leaves remainder 12. GCD(66,18) = GCD(18,12). 18 divided by 12 leaves remainder 6, so GCD(18,12) = GCD(12,6). Now 12 divided by 6 leaves no remainder, so 6 is the GCD of the original two numbers (and indeed any pair of numbers from our collection). Since long division is much easier than finding prime factors of big numbers, this method is really useful when you have big numbers to deal with. 6. a) Is it true that if GCD(a,b) = 1 and GCD(a,c) = 1, then GCD(b,c) must equal 1? b) Is it true that if GCD(a,b) = 1 and GCD(a,c) = 1, then GCD(a,b c) must equal 1? c) Is it true that if GCD(a,b) = 2 and GCD(a,c) = 2, then GCD(a,b c) must equal 2? d) Now look again at the question of whether the number of divisors of m times the number of divisors of n is equal to the number of divisors of m times n. 7. A fun GCD activity: Imagine a pool table with pockets in the four corners. Represent it as a rectangle drawn on graph paper. Start with a ball at the bottom left corner, moving up at a 45 angle. The ball bounces off each side of the rectangle, until finally it reaches one of the corners and falls in the pocket. For example, draw a 5 by 10 pool table (width 5, height 10). The ball moves up and to the right, hits the middle of the right side of the table, moves up and to the left, and falls into the top left pocket. a) How many times does the ball bounce on a 6 by 10 table? Which pocket does it land in? b) Draw tables of various sizes and make a chart listing the dimensions of the table, how many times the ball bounces in total, how many of those bounces are on the left/right sides, how many are on the top/bottom sides, and which pocket the ball lands in. Make sure to include at least the 1 by 3, 1 by 4, 1 by 6, 3 by 5, 6 by 10, 9 by 15, 1 by 12, 2 by 12, 3 by 12, 4 by 12, 5 by 12, and 6 by 12 in your list, and as many others as you have time for. [This is one of the first activities I use for teaching special cases and organization.] c) Think about greatest common factor (and about writing and simplifying fractions). Your drawings should help you understand why some of the rows in the table give the

11 c) Find a pattern in their prime factorizations, relate it to things that you found about prime patterns. d) Using that pattern, see if you can guess the next perfect number, and maybe the one after that. e) For a much much harder challenge, prove that there are no even perfect numbers except the ones contained in this pattern. f) For a much harder challenge than that, see if you can find any odd perfect numbers. Just in case we have more time, review some Divisibility Rules at least for 2, 3, 4, 5, 9, and 11. [Here s an example that works up from hard-ish exercises toward problems, once you know the divisibility rules and why they work. And if you don t, you can find it in the Ask Dr Math web site frequently asked questions, in the divisibility rules section: a) Is 7391 divisible by 11? If not, can you change one digit to make it divisible by 11? [A good example of multiple possible answers.] b) Is divisible by 3? If not, what is the nearest number that is? c) What choice(s) for the last digit * will make divisible by 4? d) Knowing the divisibility tests you have learned, how would you check if a number is divisible by 6? 2 e) Is divisible by 6? f) Knowing the divisibility tests you have learned, how would you check if a number is divisible by 8? 3 g) Prove that your test for divisibility by 8 really works: that is, that numbers divisible by 8 pass the test, and numbers not divisible by 8 do not pass the test. h) What other numbers have divisibility tests of this type? i) Can you make a divisibility test for 27 based on the same idea as the tests for 3 and 9? Hint: 999 is divisible by 27. j) By looking at the factors of 999, what other number will have the same divisibility test 2 Some students may need the hint that 6 is 2 times 3. They might even need to do some special cases: make a list of multiples of 6, and observe what they have in common (they re all even and maybe they also notice they re all multiples of 3). And if they understand prime factorization, they can see that it works both ways (all even multiples of 3 are multiples of 6). 3 Good students will check if it s divisible by 2 and 4. Really good students will realize that it doesn t work. Really really good students will be able explain clearly why it doesn t work (because the divisibility by 4 automatically guarantees divisibility by 2, so checking it provides no new information). Really really really good students will see the pattern in 2 and 4 and realize that since 8 is 2 3 that the pattern continues, so you look at the last 3 digits. Or they ll understand the reason that the divisibility tests work, perhaps because you explained them really successfully, so the hint that 1000 is divisible by 8 will help them (because they ll see it as the same as the facts that 10 is divisible by 2 and 100 is divisible by 4).

14 divide them among my two sons, but there would be one left over. If I divided them among all my five children, but there would be two left over. Maybe I should divide them among all my seventeen children and grandchildren... but no, then there would be three left over. I suppose I might as well keep them all." a) How many camels does he have? b) Are there other possible numbers of camels? c) Generalize! [For a student, I d give lots of easier examples with just two mods, and pretty small numbers, and then ask if they can give a general rule for solving all of those problems.] 17. To follow up on the squares from a few problems ago: Instead of always having the exponent 2, and different bases, it s also very interesting to start with a certain base, and look at different exponents. a) What are the powers of 2, mod 5? b) What are the powers of 2, mod 10? c) What are the powers of 2, mod 6? How many do you have to list before you get bored? d) Experiment and explore with the patterns you notice! 18. If we have time, which I m quite sure we won t in fact I suspect that nobody will even get this far in reading the handout this would be a great time to start exploring the Euler phi function, which unifies the greatest common factor 1 ideas with the ideas of some of the previous problems here.

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