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If you ever meet someone who thinks they're hot stuff in math, ask them these three questions, in this order:
1. What is a prime number? 2. How many are there? 3. Why? If they're not dumb, they'll get the first one (any number divisible only by one and itself). If they're good, they'll get the second one, too (infinitely many). But rarely does anyone get the third. The answer to that is as follows. We must start by proving something called the fundamental theorem of arithmatic. It states: Any number can be represented as a unique product of primes. A prime may be used more than once. For example, 12 = 2 * 2 * 4. There is no other way to do it with primes, according to the fundamental theorem of arithmatic. Numbers that are already prime are simply themselves. The fundamental theroem of arithmatic really contains two claims: 1. Any number can be represented by a series of primes. 2. That series is unique. Let us prove the first claim first. As mentioned above, if the number itself is a prime, that means we are done. So, we must prove this for composite (non-prime) numbers. Let us look at the first few cases: 2 is prime. 3 is prime. 4 = 2 * 2 or 4 * 1 = 2 * 2 * 1... Say we have verified the claim up to n. We wish to verify the claim for n+1. If the number is prime, we are done. If the number is composite, that means it is divisible by one of the numbers <= n. Yet, we have already verified that all numbers up through n are the unique products of primes. Therefore, n+1 is the product of two unique series of primes.. s.s..s..s.s. .s.s.s.sssssss oh ye gods man oh no no RONALD REAGAN JUST WON A COKE SNORT RACE AGAINST FREUD IN HEAVAN!!! |
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| Posted by Reverend Tedward Q. Porktanker @ 2004-06-07 16:24:14 | |
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