There are gaps between prime numbers of any size. This has already been proven and accepted by mathematicians, but with number arrays, we can prove that the gaps repeat themselves forever. Not only are there gaps of any size, but they repeat an infinite number of times.

     If we make a number array for the number 5, we will have 30 columns. The number of columns for any array is always the prime number multiplied by all lesser prime numbers. 5 times 3 times 2 equals 30. Column 1 will always contain prime numbers in any array, but the next column to conatain prime numbers is always the column for the first prime number greater than the prime number for the array. In this case, column 7 would be the next column to contain prime numbers greater than 5.

     The prime numbers are colored blue and underlined. We can see that columns 2 through 6 can only contain composite numbers. If the first number in the column is prime, every number in the column will have that prime factor. If the first number in the column is composite, every number in the coulmn will have the same lowest prime factor. In any array, all the columns from 2 to the last column before the next prime number larger than the prime number for the array, will have no prime numbers begining with the second row.

     If we made an array for the number 101, columns 2 through 102 would be composite numbers only, except for the first row. Every number in any of these columns is evenly divisible by the lowest prime factor of the first number in the column. The array for 101 would have 2.32862 times 10 to the 38th power, number of columns in it. The number of columns containing all the prime numbers higher than 101 and all composite numbers with lowest prime factors higher than 101 would be 2.774 times 10 to the 37th power. This means that begining with the square of the number 101, which is 10201, all prime numbers higher than 10201 and all composite numbers with lowest prime factors higher than 101 are limited to 11.912603% of the infinite number line after 10201.

     It has already been proven that there are gaps between prime numbers of any size, but by looking at number arrays, we can see that for every prime number, there is an infinitely repeating gap of greater size than the prime number. For any prime number, the gap is always equal or greater than the next larger prime number minus 2. The first composite number of the gap is always the prime number multiplied by all lesser prime numbers plus 2. Any multilple of this number will be the begining of another gap of equal or greater size.

     The farther you go up the number line, the larger the gaps between prime numbers. No matter how far you go up the number line, the gaps between even numbers always stay the same.

     By looking at the first 8 columns of the array for the prime number 5, we can say that any number that is (n*30)+1 must be followed by at least 5 consecutive composite numbers. Let n be any whole number 1 or greater. Let P be a prime number, and P> be the next greater prime number. Let P! be the prime number multiplied by all lesser prime numbers. (n*P!)+1 must be followed by a number of consecutive composite numbers equal or greater than (P>)-2.

     The gaps are relatively small compared to the size of numbers where they start. When you get to 39 digit numbers, the number of prime numbers is a 37 digit number. Pick a 39 digit even number and multiply the number of primes less than 1/2 of the even number times the number of primes greater than 1/2, and you will have a 74 digit number which is the number of possible pairs of prime numbers. This is what leads most mathematicians to believe that Goldbach’s Conjecture is probably true. The problem is, the number of possible pairs does not have any mathematical connection to the number of actual pairs that equal a particular even number.

     The ratio between even numbers and pairs of prime numbers is like the race between the tortoise and the hare. The 50th even number is 100. There are 15 prime numbers less than 50, and 10 prime numbers between 50 and 100. This means that at the 50th even number, there are 150 possible pair combinations of prime numbers. 6 of the possible pairs equal 100.

     For the even number 100,000, there are 5,133 prime numbers below 50,000, and 4,459 prime numbers between 50,000 and 100,000. This gives us 22,888,047 possible combinations of pairs of prime numbers. There are 457 times as many possible pairs as there are even numbers at this point, and the farther you go, the more the possible pairs of prime numbers outnumber the even numbers. The number of combinations that actually equal 100,000 cannot exceed 4,459. The number of actual pairs that equal a particular even number is always limited to the number of prime numbers greater than 1/2 of the even number.

     There is no mathematical reason, other than the plentifullness of prime numbers at the begining of the number line, for there to always be at least one pair of prime numbers that equal each even number. Between 99,900 and 100,000, there are 8 prime numbers. 3 of them equal a prime number when subtracted from 100,000. Between 1 and 100, there are 25 prime numbers, so there’s a good chance that some of the 8 prime numbers that immediately precede 100,000 will pair up with another prime number.

     No matter how large an even number that we can physically calculate, we are still at the infinitesimal begining of the number line. What happens in the infinite part of the number line that we can never see? We have already shown that there are gaps between prime numbers of any length. This means that there are gaps between an even number and the next lesser prime number that exceed any positive whole number. If the gap between an even number and the next lesser prime number is greater than a googleplex squared, all of the prime numbers less than a googleplex squared, won’t be large enough to pair up with another prime number to equal the even number. This means that we can exclude all prime numbers, because they are all to small to pair up, and equal every even number, because the gaps exceed any positive number.