Date: Jul 8, 2013 2:47 PM Author: Ian Subject: Twin Prime Conjecture proved! My name is Ian Leonard Olsen, and this is my attempt at proving the Twin Prime Conjecture. I would appreciate any feedback from all you math wizzes:
If we start by eliminating every number divisible by 2 from the number line of integers, what we have left are the odd numbers, all of which we may consider to belong to a set of "possible twin primes", since they are all separated by two. Next we eliminate every number divisible by 3 from the number line. This will eliminate some of our possible twin primes, but every time a number divisible by 3 falls on a number also divisible by 2 (which it does every 6 consecutive numbers), there will remain a set of "possible twin primes" separated by the number divisible by both 2 and 3. Next we will eliminate every number divisible by 5 (note that we skipped 4 because it is a multiple of 2 and so would be redundant), and now we have a pattern of elimination that repeats itself every 30 numbers (a result of 2 x 3 x 5). Now whenever a number divisible by 5 falls on a number divisible by both 2 and 3, which it does once every 30 consecutive numbers, the two odd numbers separated by that number remain "possible twin primes". This process can be continued indefinitely, and after each round of elimination (next would be multiples of 7), we can stop and say, yes, we have failed to eliminate all of our possible twin primes, and there must remain an infinite number of them (now they are separated by every number which is divisible by 2, 3, 5, and 7). So if after each round of elimination (using consecutive prime factors, the next being 11), there are still an infinite number of our possible twin primes left, it stands to reason that twin prime numbers are infinite in number, because as our rounds of elimination move toward infinity, our number of possible twin primes will always remain infinite.
If we want to get an idea of how fast our list of "possible primes" actually decreases as we use more factors for elimination on the number line, we can start with 2, 3, 5, and 7, so that at every 210 numbers, we have possible twin primes (209 and 211 being the first set). Now when we eliminate all multiples of 11, the result should be that 2 out of every 11 sets of our possible twin primes are eliminated, and as we include more factors, we will eliminate from the remaining twin primes: 2 out of every 13 sets, then 2 out of every 17 sets, then 2 out of every 19, etc. So the rate of decrease in twin primes should be 9/11 x 11/13 x 15/17 x 17/19 ect., (the denominators being our consecutive prime factors)