Euclid proved that there are an infinite number of prime numbers. If you multiply any prime number by all lesser prime numbers (P!) and add the number 1, the result will not be evenly divisible by P, or any prime number less than P. There will always be a remainder of 1. P!+1, must either be prime, or have a lowest prime factor higher than P. For any positive whole number, there is always a higher prime number.

     For any prime number, there is always an infinite number of gaps equal or greater than the prime number. (n*P!)+1 must be followed by a number of consecutive composite integers equal or greater than the next greater prime number minus 2.

     Subtract 1 from any gap between prime numbers, and that is the gap between an even number and the next lesser prime number. Any prime number that is less than this, is not large enough to be a part of a Goldbach pair. Since there are always infinite gaps larger than all prime numbers, all prime numbers are too small and too few to make the lower 1/2 of a pair, for every even number.

     The clincher is the fact that the ratio between prime numbers and even numbers is 0. Prime numbers are as mathematically rare as it can possibly get. They literally all but disappear, compared to the infinite number line. They are actually all but non-existent. Goldbach’s conjecture is clearly impossible.

     The next question that came to my mind after I finished my first proof was: Did I really discover something, or is this stuff already well known by mathematicians? Perhaps Goldbach’s conjecture is too insignificant to care about. If so, why the million dollar prize?

     I started doing research at the library. I looked for everything I could find on prime number theory and Goldbach’s conjecture. Generally, prime number theories predict or estimate how many prime numbers there are, less than a given number. There are several variations, but basically, you divide a number by it’s natural logarithm to get the number of prime numbers that are less than the number. The only way to test for accuracy, is to test every number and count the primes. This is very time consuming.

     There seemed to be an open question about whether or not prime numbers appear randomly, or do they have a pattern to their appearance and spacing on the number line. It would appear that my proof answers the question. If you can make prime numbers line up in columns, than they must not be random.

     Number theory is a lot bigger field than I expected. It could take a guy with only limited spare time, two or three years to research everything on the subject. Number theory is a big deal. When banks use electronic media to communicate money transactions, they must encrypt their data to prevent losses do to fraud or computer hacking. The encryption codes are based on number theories. A lot of computer software is based on number theories.

     When a bank sends electronic data to another bank, they use what is known as a public key code. Public key codes involve 250 digit numbers that have only 2 prime factors. The sender possesses a public key code. Once he encrypts the message, not even the sender can de-code it. Only the receiver can de-code the message with his part of the key, so possession of the encryption code gives a thief no advantage. The security of this system relies on the belief that a hacker would need 10,000 years to crack the code using state of the art computers.

     Suppose a high speed computer takes 4 years to verify Goldbach’s conjecture up to 14 digit numbers. It would take 40 years to get to 15 digit numbers, and 400 years to reach 16 digit numbers. What if the nanodrive, with single molecule transistors, which is currently on the horizon, becomes available and changes the processing speed from 10,000 years, to 10 seconds? A totally new encryption system would have to be developed. Number theory plays a part in just about everything that we see, have, or do.

     I could not find anything like the ratios between composite numbers and prime numbers that I had calculated. I found the previously mentioned infinite multiplication series: [ 1/2 * 2/3 * 4/5 * 6/7 * .....(P - 1)/P] = 0. It was mentioned only as a demonstration of the Law of Infinite Series. There is an infinite number of steps or calculations, none of which will actually equal 0. The Law of Infinite Series assumes that all of the infinite calculations can be made simultaneously, and the resulting product is mathematically equivalent to 0.

     Nothing was mentioned about the elements of this series being exact mathematical representations that allow us to divide the entire number line between composite numbers with lowest prime factors equal or less than a given prime number, and the percentage of prime numbers and composite numbers with a lowest prime factor higher than a given prime number. None of the prime number theories I saw were exact in any way. They all just estimate haw many prime numbers there are up to a given number. Some are more accurate in different ranges, than others. All become less accurate, the larger the number, but Mankind’s ability to calculate is limited to very small numbers. The last time I checked, they had a complete list of the first 2 billion prime numbers. This only brings us up to 11 digit numbers. What about 11 trillion digit numbers?

     I thought that maybe I could save some time by just showing my theory on Goldbach to a math teacher. There is a prestigious private college campus a few blocks away and it was summer time when school is out, so I called the head of the math department and asked him if he would be interested in seeing my theory. He said "Sure, come on over, but don’t be too disappointed when I prove you wrong."