Wednesday, 16 January 2013
First Steps into Primes and Number Theory
Sometimes it takes several tries before ideas find purchase in my hard skull. This blog started with a post by Ben Vitale that relates the number eight to the digital root of twin primes. I posted it as a number curiosity on the eighth day of the year.
5 * 7 = 35, 3 + 5 = 8
11 * 13 = 143, 1 + 4 + 3 = 8
17 * 19 = 323, 3 + 2 + 3 = 8
29 * 31 = 899, 8 + 9 + 9 = 26, 2 + 6 = 8
41 * 43 = 1763 1 + 7 + 6 + 3 = 17, 1 + 7 = 8
59 * 61 = 3599, 3 + 5 + 9 + 9 = 26, 2 + 6 = 8
71 * 73 = 5183, 5 + 1 + 8 + 3 = 17, 1 + 7 = 8
101 * 103 = 10403, 1 + 0 + 4 + 0 + 3 = 8
(Ben deserves credit for stimulating my curiosity on numerous occasions, so check out his blog site here)
A couple of days later one of my Kiwi readers, Don S McDonald, proved that the proof was so trivial it could be tweeted.... by tweeting it.
"twin primes product (6n -1)*(6n +1) =36*n sq -1 =8 mod 9 digital root."
Did I suddenly think, "Oh Yeah, this would make a great blog for introducing young students to primes and number theory?". Nope, just sent Don a quick response and back into my mental haze.
Then David Brooks wrote to add that one might explore the digital root of cousin and sexy primes, pointing out a couple of good websites to finally shake me out of the doldrums.
(And at last he finally gets to the point of the blog)
The reason I think this would be a great point to start young mathematical minds into proofs and number theory is because all the ideas in the rest of the post need only these mathematical foundations:
A) A very basic understanding of modular arithmetic topics, which is often mentioned in middle school as "clock arithmetic" and continues to be an important mathematical concept as the student progresses to more advanced math.
B) The binomial Distribution
There is a very wide range of mathematical talent in the folks who follow this blog, and many of the mathematical professors and professionals will find this post very trivial. But I am writing this with young people like my grand-nephew Alex in mind. Alex is a clever middle school student with an affinity for math and science but I fear his education is leaving him with the idea that math is about memorizing the techniques in order to answer questions quickly. I would hope he finds that math can be more exciting when you focus on the next question rather than the last answer. So for introductory math students like Alex, and the folks who teach them, a sequence of ideas to introduce number theory.
Ideas Related to twin primes
A big idea that Bob glossed over in his twitter-proof is that all primes greater than three must be one more or one less than a multiple of six. Students might be more interested in being given a question something like, "Three, five and seven are all three prime. Is there another triplet of primes of the form p, p+2, and p+4?"
For the student with even a modest understanding of modular arithmetic, the answer proof is simple. One of the three must have a factor of three.
It would seem that with a little prodding, they might be able to dig out confirmation for the 6n +/-1 model for all primes greater than 3.
At this point I might point out that two primes which differ by four are called prime cousins. Examples are (3,7), (7,11), (13,17) and (19,23). I would recast the
last question about twin primes; is it possible to have a triple of prime cousins, p, p+4, and p+6.
Students should quickly find that there is, (3,7,11). We point out that there was one set of prime triplets also, and ask, "Are there any more triples of cousins?"
Using the same modular approach from the triple of primes, they should see that there are no others.
"So, do you think anything interesting might happen if we found the digital root of the product of cousin primes?"
After a little calculation (teachers might call this practice) they will find that 3x7 = 21 has digit root of 3, but 7x11=77; 13x17=221; and 19x23= 437 all have a digital root of 5. Can this be true for all the rest.
Now we get a chance to emulate Don's binomial approach. Using (6n+1) for the smaller, the larger would be 6n+5. When we multiply we get 36n2 + 36n + 5; and the result is complete.
Then I would proceed to sexy primes. I would point out to the class that the "sexy" in this case is related to the Latin for six. I would point it out, but I doubt it would stem the flow of jokes.
Sexy primes are pairs of primes which differ by six. Five and eleven, seven and thirteen, eleven and seventeen are all sexy prime pairs.
If we return to the idea of the digital root of the product, could we find similar relation to the twin primes relation from Ben Vitale? A little experimentation leads us to 5x11 = 55, 5+5=10, and 1+0=1. Likewise we note that 7x13=91 also with a digital root of 1. Students are prone to jump to conclusions (teachers are too) and many may call out that answer.
If they do, just ask them if they can show that is true for all the values algebraically. Others who may have gone on to 11x17=187 will realize that the prediction is false. Now we can encourage them to press on and look for a pattern they think is true. Eventually one will be able to show that the digital root is always 1, 4, or 7 or of the form (3n+1).
By leading them to look at the alternate form of the statement, that all products are congruent to 1 mod 3, the problem can be reduced to two simple cases.
Hopefully we can lead them to figure out that numbers of the form 6n+1 are congruent to 1 mod 3, and 6n-1 forms would be congruent to 2. Now we need them to discover that both the sexy primes must be of the same form, either 6n+1, or 6n-1.
If both numbers are 6n+1, the product must be a number that is equal to 1 mod 3. If both are of the form 6n-1, the product must be a number that is equal to 2x2= 4 mod 3, which is also 1.
After no success finding triplets of primes, or prime cousins, students may expect that the same result may occur with sexy primes, but they should discover several triples that are less than 100; (7, 13, 19) and (17, 23, 29) both show up quickly. In fact, there are sexy prime quadruplets such as (11, 17, 23, 29) that they might discover as they search for triples. There is an interesting pattern in the sexy quadruples that might tease students interest, all the first primes in the quadruples end in one.
There is even one sexy quintuplet, (5, 11, 17, 23, 29), but since every fifth number in such a sequence is divisible by five, there can be no more.
It may even be worth pointing out to younger students that there is a general name for numbers which differ by a common amount. They are called arithmetic sequences.
And as a reward, you might try this puzzle by David Wells from his "The Penguin Book of Curious and Interesting Puzzles" which can be solved with some clever use of the idea of modular congruence. I'll post the answer down the page a ways. Enjoy..
An old invoice showed that seventy-two turkeys had been purchased for "x67.9x". The first and last digits were illegible. How much did each turkey cost?
Since 72 is divisible by both 9 and 8, we know that the number must have a digital root of zero, and the last three digits will be divisible by 8 (equal to 0 mod 8)
So 79x must be divisible by 8, and the last digit must be a 2.
Now we can use the fact that the digital root must equal 9 (or 0). 6+7+9+2 has a digital root of 6, so we need three more to make nine, and 3 must be the first digit. Now all we need to do is take the total price, 367.92 and divide by 72. It appears that each bird costs 5.11 (Pounds, dollars, Euros, or other monetary units of your choice