Since it came up, I thought I'd show you the quick proof that there is no greatest prime number. This type of proof is called a reductio ad absurdum. That means it works by showing that a denial of the thing to be proved leads to a contradiction.
We assume (for the moment) that there is a greatest prime number-- call it n. This means the list of prime numbers is finite, and looks something like this:
{2,3,5,7,...,n}.
Now, we multiply all of those numbers together, and add 1 to the result. This gives us a new number P.
P= (2*3*5*7*...*n) + 1.
What do we know about P? P is not divisible by any of the prime numbers on our list-- they all give a remainder of 1. There are two possibilities. P might be prime. But in this case, P is a prime greater than n, which contradicts our initial assumption. Or, P isn't prime. In this case, P is factorizable-- and by extention, P is prime-factorizable. But because our list of primes was supposed to be complete (up to n), P's prime factors won't be on the list-- they must be greater than n. Again, this contradicts our assumption. And this exhausts all the possibilities.
Our conclusion? The initial assumption was false. We can't name the greatest prime number-- because there isn't one. That's it.
(No more math. I (almost) promise.)
1 comment:
makes sense. i want to talk more about the unicorn and Portuguese. because unicorns are the result of mental formations, obviously (if i can use the word) not based on reality, the concept of unicorn malleable and subject to change. i would suppose it went something like this (obviously, this is only speculation). caveman A says, "wouldn't it be cool if a horse had a horn, somewhere in the woods?" caveman B says "based on what i know,lots of animals have horns, this is certainly possible. lets say it exists. we'll call it a unicorn." so the form of unicorn was created (in their minds at least.) they then tell caveman C about the idea of unicorn. caveman C says "well, unicorns, with that horn, must be magical." caveman A and B agree, and now the form of unicorn has changed. then, in our class, caveman D (just kidding) says to us, " all unicorns, with their magical horns, are definitely Portuguese. speaking Portuguese is an essential part to what a unicorn is, just as important as the horn." and if everyone agrees, then the form of unicorn has changed again. what was a unicorn before is no longer a unicorn. there is nothing to hold the form solid.
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