[Now that the 'Carnival is Over' (is that another song title?), it's time to return to the essence of this blog.]
There is no end to the number of articles one can find on the internet and in the literature regarding prime numbers, from famous theorems to unsolved problems that seduce budding young mathematicians.
The following investigation is intended for middle school students, working in research teams, but can be extended to secondary students who want to explore mathematics further in their classroom or in their Math Club.
Student Investigation
Notice that
4 = 2+2
6 = 3+3
8= 3+5
10 = 3+7 = 5+5
The purpose of this investigation is to explore part of the world of prime numbers and become a mathematical researcher. Mathematicians, like scientists, observe phenomena, look for patterns, make conjectures and generalizations and try to prove them. Mathematicians seek to understand the secrets (general truths) underlying patterns and relationships in numbers and shapes.
1. Based on the first few examples above, do you think your mathematical research team has enough information to make a conjecture, or educated guess, about even positive integers? By the way, why didn't we begin the pattern from the first even positive integer, 2?
2. Let's continue the exploration. Begin by making a list of all primes up to 100. Why does it make sense to have this list available?
3. A table is very useful to organize your data and form hypotheses. A suggested table is provided below. One of the column headings needs to be completed. Then complete the table for even integers up to 30. Your team leader should assign a few of these to each member of the team.
Even Positive Integer........Number of Ways to ___............List of ways
4..................................................1........................................2+2
6
8
10...............................................2........................................3+7;5+5
12...............................................1.........................................5+7
.
.
.
30
4. Here's an example of a conjecture:
Based on
22 = 3+19; 5+17; 11+11
24 = 5+19; 7+17; 11+13
26 = 3+23; 7+19; 13+13
28 = 5+23; 11+17
one might conjecture that even numbers can be written as a sum of two primes in at most three ways.
Do you think it's easier to prove this 'educated guess' or disprove it? Try it! You may need to extend your table!
5. Extend your table to even positive integers up to and including 60.
6. Based on this table, your research team now has to make at least three conjectures, then attempt to disprove them or provide an explanation for why they may be true. You may need to go beyond your table.
7. Jeremy determined that
60 = 7+53; 13+47; 17+43; 19+41; 23+37; 29+31 and
100 = 3+97; 11+89; 17+83; 29+71; 41+59; 47+53
He conjectured that six is the greatest possible number of ways that an even number up to 100 can be written as a sum of two primes. Disprove it! Again, you might need to extend your table.
8. Is your research complete? Do you think a mathematician would make other conjectures about even numbers or think of other problems related to sums of primes? Perhaps, numbers that can be written as a sum of three primes? Sums of consecutive primes like 3+5+7. Perhaps you'd like to continue....
9. [When the activity is complete] Research Goldbach's Conjecture on the web and write a brief description of its history. Has it been proved?
As usual, make suggestions for improving this; revise, edit, enjoy...
If you use this in the classroom, please share the experience. The feedback is invaluable to me.
Monday, September 24, 2007
Putting the Gold 'Bach' into Primes! An Investigation...
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11 comments:
Dave, this investigation looks great, as always. :)
One issue I have though is that I don't think there's enough "leading" at the beginning to get them to come up with the kind of conjecture that you want. Based on your examples, I think most of my students would "conjecture" that all even numbers can be represented as the sum of two odd numbers. And oh, so much easier to prove. ;-)
I would add something like:
Today we are going to examine the different ways numbers may be represented as the sum of two (or more) prime numbers. Remember that 1 is not a prime number, and that 2 is the only even prime.
In addition to wanting to try this with my middle schoolers, I have a pair of 9yos I think I'd like to give this to as something to pull out and work on over time when they finish their other math work ahead of the rest of the class. They've both already shown a lot of interest in prime numbers and trying to find patterns related to them.
With my middle schoolers, I'm currently following a trail from Gauss sums to triangle and other figural numbers, to handshake problems, so I won't get to this soon, but I'll try to come back and tell you how it goes when I do try it. Also if I give it to the younger guys, I'll let you know what they think.
Great suggestions, mathmom. Your wording is clear and will get them started in the right direction more quickly.
However, I've worked with some middle school math groups and the discussion that might result from intentionally leaving out details with guidance by the instructor can be of value. Yes, some will jump to the pattern of odds you mentioned but others will challenge that with with the 4 = 2+2 example and why 8 = 1+7 or 10 = 1+9 is missing from the list. Sometimes I intentionally leave it vague like this and make it part of the challenge. Try it with your own sons or some of the older students and see what happens. This is particularly effective when one has mixed ability groupings. Students will correct each other and the end result will be fine. Let me know!
One way to extend it is to explore the numbers that are the sum of three primes.
In the past, I read sections of The Number Devil to my students and we did some similar work. It wasn't as much of an investigation, but they were still interested. We did the three primes I mentioned above as well.
I think I'll try this with one of my math sections next week. Thanks for reminding me of it and presenting it as more of an investigation.
Ah, Lynx. Vinogradov's theorem: every sufficiently large odd number is the sum of three primes. MathWorld points out that the best value of "sufficiently large is about 3.33×10^{43000}.
There's also Chen's Theorem that every sufficiently large even number is expressible as one or both of p+q or p+qr, with p, q, r prime.
The lesson for students should be the importance of the 'sufficiently large' clause; it shows up in many places in mathematics. Consider Waring's problem: for each positive integer n≥2, there is a number g(n) such that every integer is the sum of at most g(n) nth-powers. every integer is the sum of at most 4 squares, at most 9 cubers, and at most 19 4th powers.
The hard problem is to find G(n) such that every sufficiently large integer is the sum of at most G(n) nth-powers. It turns out that G(3)≤7. It may be less than 7.
The other lesson is to teach the idea of arithmetic density. There are more than one way to measure the size of a set. The even and the odd numbers both have density ½. The primes have density 0; so do the squares. But they are all infinite sets. So, what's the important measure of size for these sets? Both are important in different circumstances.
And then there's logarithmic density. The primes have log density 1. The squares have log density ½. It turns out that you expect a (modified) Goldbach or Waring-type result for any set with positive log density.
Feel free to use any of the problems at my site...
http://www.unsolvedproblems.org/
Tim
lynx--
The Number Devil sounds engaging. Tell us more about it, its intended age range (kids of all ages?)and a brief description. I found it at Amazon but I haven't seen this book before.
As far as turning rich engaging mathematics into full-blown classroom investigations, well, that's why I write this blog! Come back often and, if you use it in class, let me know how students reacted and how you needed to modify it for them. Thanks!
eric--
What can I say. You are our resident expert who brings so much more to what I write. Arithmetic density was a new phrase for me, a bit different from density per se. Your wealth of knowledge is awesome.
timro21--
Nice website. I may add that to my blogroll and recommend the list of unsolved problems to the educators who stop and visit here. Thanks for sharing. I hope you enjoyed my little student activity. I delight in bringing the mysteries of primes and number theory to a larger audience.
Arithmetic density is just density--I added the adjective to distinguish it from log density further down. Take an increasing sequence a_k, and its counting function a(x) = #{ a_k | a_k ≤ x}. The counting function for the primes is π(x), for example. Then, the density is lim_{n→∞} a(n)/n.
So, the density of the even numbers is 1/2, as approximately n/2 of the positive integers ≤ n are even. π(n)/n ≈ 1/log n, so the density is zero.
Log density is the limit of log a(n) / n. That the squares have log density ½ makes the results that every prime ≡ 1 (mod 4) is a sum of two squares, every number not of the form 4^k (8n + 7) the sum of three, and every number be the sum of 4 plausible. Yet, Log density is unaffected by multiplying the entire sequence by a constant. What sort of results would you have for the sequence of numbers like 2n²? That's why I said 'modified' above.
This type of measure doesn't sum the way areas and volumes sum, exactly. The integers have log density 1; so do the primes. So do the sums of two squares. They're different sets.
The Number Devil -- this is a great book, and I would say good for all ages (including adults). It has large print, color illustrations, and a fiction story line that makes it appealing even to fairly young kids (and a great choice for young gifted kids) but the mathematical content is great for middle/high school level kids, and fun for curious adults as well, especially those who may not have been exposed to all the "fun" math stuff in the past.
thanks, mathmom, for the excellent review/recommendation!
makes me want to run to the library, being cheap and all that!
dave
Another point about this investigation. While the focus has been on primes, I hope my readers recognize that I had a few other objectives here. Exposing students to the nature of math research -- the whole notion of the process: posing questions, tackling a problem, collecting data, looking for patterns/relationships, making conjectures, generalizing/abstracting, attempting a proof or posing other questions that may lead in a different direction,...
I've always believed that this process is similar to the Scientific Method and should be explained as such to mathematics students. And the ost important personal quality for mathematical research? I suggest persistence in the face of frustration! Just ask Prof. Wiles, or, better yet, his wife!
Thanks for another great blog. This morning I was re-reading it with the intention of commending your emphasis on the process and on asking questions, but I see you've just preceded me in pointing this out :).
I believe the results we get in math, in science, or elsewhere for that matter, depend a whole lot on the process followed and the questions asked. Glad to see you're using math as a vehicle to expose students to this.
Marc
More about The Number Devil.
I used it at the end of the year last year with my 5th graders. It was that last week of school when you can't really do anything, but I still wanted them to be engaged and doing stuff. Each day during math time, I read them a chapter and then we did some work with the concepts in that chapter. We didn't make it through the whole book (by design, the later chapters get a bit beyond what my 5th graders were ready for). They really enjoyed the quick-tempered number devil. Other topics explored: primes, Pascal's triangle, infinite series. Lots more too, but the book is at school and I'm at home. I highly recommend this book. I bought it for myself, but immediately saw the applications in the classroom. Many of my kids didn't realize they were still 'doing math' when we read and did activities with the book.
I'll definitely get back to you next week after the investigation with my advanced math section (7th graders).
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