Now we know the exact ratio between prime numbers and composite numbers for the whole number line. The ratio is 0. This makes prime numbers relatively rare. 0 is a pretty low number. We see 25% prime numbers in the first 100 integers of the number line. That is because all of the prime numbers are at the beginning of the number line, and the beginning of the number line is all that we can see. No matter how large or numerous the integers we calculate, we can never go beyond the infinitesimally small beginning.

     There is another infinite series equation that verifies these ratios between prime and consecutive numbers. My word processor cannot display the proper symbols that mathematicians like in their equations, but I’ll try to get them to understand it anyway. [ 1/2 * 2/3 * 4/5 * 6/7 * 10/11 * ........ (P-1)/P ] = 0.

     I found this equation in textbooks on number theory, but the only thing that was said about it, was that it equals 0. It is used to demonstrates the law of infinite series. I imagine that the first mathematicians to work this equation were using slate tablets. Naturally, they would reduce each element to its lowest common denominator. But if you look at each element as a whole number ratio, instead of viewing it as a fraction, and you don’t reduce each element, you get exact ratios between prime and consecutive numbers.

     The first element is 1/2 for the number 2.. For any 2 consecutive numbers, 1 of them will have a lowest prime factor higher than 2. The next element is for the number 3. 1/2 * 2/3 = 2/6. Out of any 6 consecutive integers, 2 of them will have lowest prime factors higher than 3. The element for the number 5 is: 2/6 * 4/5 = 8/30. For any 30 consecutive integers, 8 of them will have lowest prime factors higher than 5. The next element is for 7. 8/30 * 6/7 = 48/210. For any 210 consecutive integers, 48 of them have lowest prime factors higher than 7.

     If you make number arrays for any of these elements, you will see that all of the higher prime numbers line up in columns. If all the prime numbers higher than the prime number element are restricted to the ratio or percentage the element represents, and the equation converges to 0, then that is the ratio between prime numbers and composite numbers for the whole number line. There is also another way to show it with an infinite number array:

2:         4,       6,        8,     10,   12,    14,     16,    18,      20,     22,......

3:         9,    15,     21,     27,    33,    39,     45,     51,      57,     63,......

5:       25,    35,    55,     65,    85,    95,   115,   125,   145,    155,......

7:       49,    77,    91,   119,  133,  161,  203,   217,    259,    287,......

11:   121,  143,  187,  209,  253,  319,  341,   407,    451,    473,......

13:   169,  221,  247,  299,  377,  403,  481,  533,    559,    611,......

17:   289,  323,  391,  493,  527,  629,  697,  731,    799,    901,......

19:   361,  437,  551,  589,  703,  779,  817,  893,  1007,  1121,......

     There are an infinite number of columns and an infinite number of rows. All of the prime numbers are in the first column. All of the rest of the columns are composite numbers. Each row contains all of the numbers that have the prime number as a lowest prime factor. Each prime number has it’s own separate set of composites. No number can appear more than once in this array. For each prime number, there is a separate infinity of composite numbers. The ratio between prime numbers and composite numbers is equal to 1/infinity, which is mathematically equivalent to 0. Infinity divides into 1 exactly 0 times.

     Even numbers are 50% of the number line. Prime numbers are 0% of the number line. Are there enough prime numbers to make a pair for every even number? Let’s consider the even numbers that are also evenly divisible by 4. These are 25% of the number line. For each pair of prime numbers, one must be less than 1/2, and one must be greater than 1/2 of the even number. For the number 100, there are 14 prime numbers that are less than 1/2, and 10 prime numbers that are greater than 1/2 but less than 100. There are 140 possible pairings, but only 6 pairs will equal 100.

     The number of possible pairings is equal to the number of prime numbers in the upper 1/2 times the number of prime numbers in the lower 1/2. The number of possible pairings have nothing to do with the number of pairs that actually equal the even number. The number 104 has 168 possible pairs, but only 5 pairs that equal 104.

     As the even numbers get larger and larger, there are more and more possible pairs. But the possible pairs have no mathematical connection to the actual pairs that equal the even number. Some even numbers have lots of prime pairs, and some have very few. It’s just that, at the begining of the number line, prime numbers are so plentiful that you can find at least one pair for every even number that you calculate.

     No matter how far you calculate greater than the number 1, the density of even numbers never changes. It’s always 50%, but the density of prime numbers does change. It reduces to 0%. For any arbitrarily small fraction, there is a point on the number line where the percentage of prime numbers per consecutive numbers is less than the arbitrarily small fraction.

     At what rate does the percentage of prime numbers decrease? In the infinite series equation we looked at, [ 1/2 * 2/3 * 4/5 * 6/7 * 10/11 *....(P-1)/P] = 0, each element in the series tells us what the percentage of remaining prime numbers per consecutive nembers is less than, starting at the square of the prime. For any prime number, the first composite number to have that prime number as a lowest prime factor is always the square of the prime. 2 squared equals 4. Beginning with the number 4, all prime numbers are limited to less than 1/2 of the number line. 1/2 * 2/3 = 1/3. 3 squared equals 9. Beginning with the number 9, all prime numbers are less than 1/3 of the number line. 1/3 * 4/5 = 4/15. 5 squared equals 25. Beginning with the number 25, all prime numbers are less than 4/15 of the number line.

     I worked this equation for the first 100,000 prime numbers. I made an arbitrary chart of whole number reciprocals for column 1. I then attached the prime number who’s element was the first one less than the whole number reciprocal, for column 2. Column 3 represents each prime number in column 2, divided by the prime number preceeding it. Here is the result:

Column 1            Column 2                          Column 3

Reciprocal        Prime Number      Prime Number/Previous Prime

  1/2                         2

  1/3                         3                            1.5

  1/4                         7                            2.333...

  1/5                        13                           1.857142857

  1/6                        23                           1.769230769

  1/7                        43                           1.869565217

  1/8                        79                           1.837209302

  1/9                      149                           1.886075949

  1/10                    257                           1.724832215

  1/11                    461                           1.793774319

  1/12                    821                           1.780911063

  1/13                  1451                           1.767356882

  1/14                  2549                           1.756719504

  1/15                  4483                           1.758728913

  1/16                  7879                           1.757528441

  1/17                13859                           1.758979566

  1/18                24247                           1.74954903

  1/19                42683                           1.760341486

  1/20                75037                           1.758006701

  1/21              131707                           1.755227421

  1/22              230773                           1.752169589

  1/23              405401                           1.756708974

  1/24              710569                           1.752755913

  1/25              1246379                         1.754057664

     The first column is a number line of reciprocals. The second column is the first prime number who’s element is less than the reciprocal. The third column is the prime number divided by the previous prime number. We can see that it converges to the number 1 and 75/100. The fluctuation is limited to a few 1/1000 after that, but all I have is a pocket calculator. To go beyond the first 100,000 prime numbers is just not practical without a computer.

     We can use this 1.75 number to predict where the density of prime numbers decreases. 1/25 equals 4%. The first prime number who’s element is less than 4% is 1,246,379. It’s square is 1,553,460,611,641. After this prime square, all higher prime numbers and all composite numbers who’s lowest prime factor is 1,246,379 or greater, are limited to less than 4% of the rest of the number line. After this prime square, a little more than 96% of the rest of the number line is composite numbers who’s lowest prime factor is less than 1,246,379. If we multiply this number by 1.754 to the 75th exponent and then square the result, we can get a close prediction of the point where the remaining prime numbers are less than 1/100 of the numberline. Multiply by 1.754 to the 975th exponent and then square the result, and we can predict the point where the remaining prime numbers are less than 1/1000 of the remaining numberline.

     A large University with state of the art computers could work the infinite series equation for the first 2 billion prime numbers and see where the 1.75 number converges to. A pocket calculator does not handle very large numbers too well, but I found that there is a correlation between the percentage, and the number of digits in the number. To get to where the remaining prime numbers are less than 1/100, you must have a 49 digit number. To get to where they are less than 1/200 you need a 98 digit number. To get to where they are less than 1/1000 you need a 490 digit number.

     At approximatly 490 digits, 99.9% of the remaining number line is composite numbers. At 4,900 digits, the composite numbers are 99.99% of the remaining number line. A 4,900 digit number is far beyond what mankind can calculate as far as Goldbach’s conjecture is concerned. The first even number that cannot be the sum of two primes will probably always be a mystery. But it is definitely out there. The percentage of prime numbers forever drops toward 0%. The first even number that can only be the sum of four primes may need to have trillions and zillions of digits in it. Typed in number 8 pica, the number may stretch from here to the next galaxy.