Prime Numbers
Or the Prima Donnas as Hans Enzensberger calls them because they are very special. His little book of fiction, The Number Devil: A Mathematical Adventure is a beautifully written, treasure trove of little number essays written around a boy who daydreams through twelve nights. (The KIS Grade 6 math teacher has a copy to borrow if you are interested).

Sieve of Eratosthenes

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How about a few challenges to try out?
Now that you know what Prime Numbers are and can find a few, here are
three followup puzzles for you to work out. These problems are not usually found in MS or HS textbooks. I will be
relying quite a
lot on various examples used in The Number Devil book mentioned above.
1. Pick any number larger than 1. Multiply it by 2. See if you can find a prime number between the first and the second number. An example: Take 20. Multiply by 2 to get 40. Can you find a prime number in between 20 and 40? 2. Take any even number larger than 2. Can you find two primes that add up to that number? An example: 10 = 3 + 7 How many can you find? Can you find any that do not work? 3. Take any number larger than 5. Can you find 3 prime numbers that add up to your number? An example: 38 = 2 + 17 + 19 How many can you find? Can you find any that do not work? These seemingly simple problems such as these have severely challenged the best minds of the planet over many years. Many of the patterns have resisted proof and still live on as Conjectures. They seem to always work, but if you can find one counterexample that does not work, that would be a great find. We'll hear much more from The Number Devil wonder book of number treasures throughout these web pages. This particular area of mathematics is usually called Number Theory, the applications of which have many practical uses. One such use is the RSA algorithm (Rivest, Shamir, Adelman the inventors  see their photographs above) of a special encryption system for securing banking transactions and other sensitive electronic data transmissions that rely on factoring two very large prime numbers. The ability to break up any composite number into its unique set of prime factors is called the Fundamental Theorem of Arithmetic. Thus, these questions arise: How do we know if a number is prime or composite? And, especially when presented with a huge number where the Sieve of Eratosthenes is not useful? Take the two numbers below: 1a. How can you tell if this number is prime or not (composite)? 10 000 019 1b. How about an even bigger number? 141 421 356 237 307 Hint: You may need to use your rules for divisibility to try to work it out. Or maybe even write a simple computer program, if you have that knowledge. Or use a special prime number tester here. (Scroll down the page just a little). As you can see there, they claim that all prime numbers can be written as either 6n +1 or 6n  1. The website authors also provide a nice prime number puzzle to try out on your friends there too. OK, so now you may want to know the largest prime number found so far? It can be seen here. The 45th Mersenne prime was found on on 4 October 2009 and earned a $100,000 prize for UCLA and others. (Scroll down a little to see the result, almost 13 million digits. And notice the huge exponent of 2 combined with a 1 at the end)  2^43,112,609 1. A description of a Mersenne Prime (not just any old prime) is given at this website buried down on the Research Lab page.: " ... Mersenne primes have to fit the formula (2^n 1). [n must also be a prime number too.] By the way, not all numbers that fit the formula are primes, just some of them. So a NORMAL prime can be any prime number, but to be a Mersenne prime when you add 1 it must make a power of 2.
If interested, a nice history of primes is given here, separated into before computers and after computers historical periods. For instance, "by 1772 Euler had used clever reasoning and trial division to show 2^31 1 = 2 147 483 647 is prime." And here is a nice applet that finds prime factors for you up to 2^61 1 and, thus, also checks for prime numbers.
