PFS 34-6 2/24/20 ________________
Prime Numbers
Choose one project that’s an appropriate challenge for you, and do a good job with it! If you
have time after that, you're welcome to work on other parts of the POW. Try to work with other
students, family members, friends, and teachers.
We can write any word in English by combining some of the 26 letters of the English
alphabet. And in science, all compounds are made up of some of the 118 elements. In math,
prime numbers play a similar role, as every whole number can be built up from prime numbers.
Prime numbers are numbers with exactly two factors. Here's a way to picture this. If we
have 13 square tiles, the only rectangle we can make from them is a “snake,” 1 x 13. But if we
have 12 tiles, we can make 1 x 12, 2 x 6, and 3 x 4. Thirteen is a prime number, and its factors
are 1 and 13. Twelve is a composite number, and its factors are 1, 2, 3, 4, 6, and 12. The
number one is a "unit," and all greater numbers are prime or composite.
People have been studying primes for over two thousand years. We've learned some
things about them, but there are still many mysteries!
1) Make a neat chart of the first 20 (or more) counting numbers. Label each number as a unit, a
prime, or a composite. For the composites, write their factors. Your chart should look something
like this:
1 Unit
2 Prime
3 Prime
4 Composite 1x4,2x2
5 Prime
6 Composite 1x6,2x3
After you complete your chart, write down anything you notice. Do the prime numbers seem to
show up in a regular pattern?
2) In 1742 Christian Goldbach wrote a letter to the great mathematician Leonhard Euler,
wondering if it was true that every even number greater than 2 could be written as the sum of
two prime numbers. This is an unsolved problem! No one has ever found an even number that
couldn't be written this way, but no one has found a proof that it will always work. Do some work
on Goldbach's Conjecture. Make a chart starting like this:
4=2+2
6=3+3
8=3+5
10 = 3 + 7 = 5 + 5
Write down anything you notice!
