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Math Help > Number Theory > Prime
The concept of the "Prime" number goes back thousands of years, and forms the basis of Number Theory.
"Prime" means it can't be factored -- of a whole number, it means it can't be expressed as the product of smaller numbers. 12 is not prime, because it can be expressed as 2 × 6. 41 is prime, because it can't be expressed as the product of smaller numbers. Of an algebraic expression, "prime" means it can't be expressed as the product of two or more simpler algebraic expressions.
The Sieve of Eratosthenes is a simple method to make a list of prime numbers.
Prime Factorization means expressing a number as the product of prime numbers. Did you know there's only one way to do this for any given number? Euclid proved that two thousand years ago!
The Greatest Common Divisor (GCD) of two or more numbers is the largest number that evenly divides all the numbers.
The Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of all the numbers. It's hard to find the LCM of two numbers, but this page has a trick that might help you!
Counting Primes is all about proving that C(n,k) is an integer, as is more complex expressions involving factorials in the numerator and denominator of a fraction.
The sum of the Reciprocals of Primes does not converge. Interestingly, the sum of reciprocals of primes that don't contain "2004" in their decimal representations does converge, offering a fascinating proof that an infinite number of primes contain "2004" in their digits.
More advanced theorems about GCD etc.
The webmaster and author of this Math Help site is Graeme McRae.