Prime Numbers: the atoms of mathematics (March 25, 26,30, 31)

The Atoms of Mathematics 

Prime numbers are the atoms of mathematics. All numbers are made up of factors of prime numbers, 1 and/or composite numbers., those numbers divisible by exactly two distinct factors: 1 and the number itself. We discussed the history of the number one and its prime designation for thousands of years until the last century when it was determined that it was neither prime nor composite. Composite numbers are divisible by some prime number such as 2, 3, 5, 7, 11 and so on. I used tennis balls to illustrate that if you can make a rectangle of 2 x ____ or 3 x ____ and so on, the number is not a prime number, but composite. If the number of tennis balls can make only a 1 x ___ rectangle like the number 2 or 3 or 5 or 7, it is prime.

Curious Incident of the Dog in the Night-Time

I shared with the children the card I received while attending a play in London called The Curious Incident of the Dog in the Night-Time. Christopher, the lead actor loves mathematics and especially prime numbers. If you were seated in a prime number seat, you received the Curious Badge challenge. If your full name is a prime number when using the ascending dollar word value (A=1, B=2, …Z=26), you would receive a Curious Badge. Since my name, David Kramer was worth 106, not a prime number, I did not receive my badge. I challenged the children to try their names and all of their friends and family names. They can use the attached pdf to help them calculate the dollar word value.

The Sieve of Eratosthenes 3rd Century BC

I then introduced them to the ancient Greek method of finding prime numbers. This method called the Sieve of Eratosthenes, is still used today with supercomputers. Essentially, we started with all of the natural numbers from 1 to 300 in rows of 10. As the 3rd century BC mathematician, Eratosthenes, used a metaphorical sieve to shake out all of the composite numbers and the number one so only the prime numbers would emerge. The first step is to cross out the number one since it is neither prime nor composite. Then circle the first five prime numbers because they are easy and somewhat obvious: 2, 3, 5, 7, and 11. 

Eliminate Multiples of 2 and 5 — Vertical Lines

Then we can eliminate almost half of the numbers by vertically crossing out all even numbers except for 2: the numbers that end in 2, 4, 6, 8, and 0. Then we cross out the numbers that are multiples of 5 by drawing a vertical line through all numbers ending in 5 except for 5. The numbers ending in 0 are already crossed out since they are also even.

Eliminate Multiples of 3 and 11— Diagonal Lines

Then the challenge begins by eliminating all multiples of 3, except for 3. The pattern is quite beautiful as a simple forward slash diagonal. Many of the multiples of 3 were already eliminated because they were also multiplies of 2 and 5 such as 6, 12, 15, 18, etc. but it would pick up new multiples such as 9, 21, and 27. There is an easy method of testing whether a number is a multiple of 3 by adding the digits. if the sum of the digits is a multiple of 3, then so is the original number. The next easiest prime multiple is 11 since it is a back slash diagonal that picks up 77.

Eliminate Multiples of 7 and 13 — Super Knight’s Moves

The next two prime multiples are similar: 7 and 13. The patterns are identical in that you add ten and then subtract 3 or add 3. The first 300 numbers requires that you eliminate multiples of one more prime: 17. This requires you add 20 and subtract 3 or add 10 and add 7. The attached pdf shows all of these eliminations in color so you can see each multiple separately. On the back side of the pdf is a reduced page of each multiple pattern. When the 7th prime multiple is completed, the numbers that have not been eliminated should be circled and are prime. There are 25 prime number in the first 100, 21 prime numbers in the second 100, and and 16 prime numbers in the third 100. The children may be curious as to how many prime numbers are in the next 100 (from 301-400) and so on.

Exploring Patterns in Primes

For over two thousand years, mathematicians have been searching for that elusive pattern that would predict the next prime number. Next week’s lesson will focus on my life-long pursuit of this pattern. I have only been looking for 47 years and am getting close. The monetary prizes are in the millions of US dollars, but more importantly, the mathematician who solves this puzzle will have cracked the greatest code in history.