Monday, September 24, 2007
Putting the Gold 'Bach' into Primes! An Investigation...
[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.
Posted by Dave Marain at 7:19 AM 11 comments
Labels: conjecture, Goldbach Conjecture, investigations, middle school math, primes