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For any 2 consecutive whole numbers, anywhere on
the number line, 1 of them will have the number 2 as a lowest prime factor. This means that exactly 1/2 of the entire number
line has the number 2 as it’s lowest prime factor. 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 number. This means that all of the prime numbers after
the number 4 are restricted to the other 1/2 of the number line.
If I make a 2 column number array, all the even numbers will line up in the 2nd column. All of
the prime numbers and their products after the number 2, will line up in column 1.
For every prime number, there is an exact ratio of the entire number line that has that prime
number for the lowest prime factor. For the number 3, the ratio is 1/6. For any 6 consecutive numbers anywhere on
the number line, exactly 1 of them has the number 3 for it’s lowest prime factor.
If we add the ratios for the numbers 2 and 3 together, (1/2 + 1/6 = 2/3), we can say that exactly
2/3 of the entire number line is the product of the number 3 or less. All prime numbers higher than 3 squared, and their products, are
restricted to the 'other' 1/3 of the number line.
If we make a 6 column number array, we can see that all of the even numbers line up in 3 of the
columns, all of the numbers that have the number 3 for the lowest prime factor, line up in the 3rd column,
and all of the prime numbers higher than 3 and their products line up in 2 of the columns, column 1 and column 5. For
any prime number, there is an exact number of integers per X number of consecutive integers that have that prime number for
the lowest prime factor. I will refer to this as the 'ratio' for the prime number.
By adding the ratio for each prime number, one at a time, and charting the series, we should be
able to see where it converges. This would give us the exact ratio between prime numbers and composite numbers for the whole
number line, without having to actually calculate the entire number line.
To find the ratio for any prime number, multiply the prime number by all lesser prime numbers.
5 * 3 * 2 = 30. Out of any 30 consecutive numbers, we know that 15 of them are even numbers. Out of the 15 remaining, 5 of
them will have 3 for their lowest prime factor. Out of the remaining 10 numbers, 2 of them will have the number 5 as a lowest
prime factor. For the prime number 5, exactly 2 out of any 30 consecutive numbers in the entire number line have the number
5 for their lowest prime factor.
If you make a 30 column array, you will see that 2 of the columns, the 5th and the 25th,
will have nothing but all of the numbers that have 5 as a lowest prime factor. All of the prime numbers higher than 5 will
line up in 8 of the columns, columns 1, 7, 11, 13, 17, 19, 23, and 29. All of the composite numbers in these columns will
have lowest prime factors of 7 or higher.
There is a short cut to finding the ratios for each prime number in sequence. To find the ratio
for the number 7, multiply 7 times the denominator in the ratio for 5. 7 times 30 equals 210. The denominator for the number
7 is 210. To find the numerator, subtract 1 from the previous prime number and multiply it by the numerator in the ratio for
the previous prime number. 5 minus 1 is 4. 4 times 2 is 8. The ratio of the entire number line that has 7 as a lowest prime
factor is 8/210. For any 210 consecutive integers, exactly 8 of them have 7 as a lowest prime factor.
To find the ratio for 11, multiply 11 times 210 to get the denominator of 2310. To find the numerator
subtract 1 from 7 to get 6, and multiply by 8 to get 48. The ratio of the entire number line that has 11 as a lowest prime
factor is 48/2310. For any 2310 consecutive numbers, exactly 48 of them have 11 as a lowest prime factor. The following chart
shows the ratios for the first ten prime numbers.
Prime Number
Ratio
2 1
/ 2
3 1
/ 6
5 2
/ 30
7 8
/ 210
11 48
/ 2310
13 480
/ 30030
17
5760 / 510510
19 92160
/ 9699690
23 1658880
/ 223092870
29 36495360
/ 6469693230
If we add all these ratios together, we get 5447823150 / 6469693230. This means that for any 6,469,693,230
consecutive numbers, exactly 5,447,823,150 are composite numbers that have 29 or less as a lowest prime factor. This means
that all of the prime numbers higher than 29 and all composite numbers with lowest prime factors higher than 29 are limited
to the other 1,021,870,080 of the 6,469,693,230 consecutive numbers.
If we convert these ratios to percentages, it makes them easier to understand. Beginning with
the square of 29, exactly 84.2052776% of the rest of the number line is composite numbers with lowest prime factors of 29
or less. All prime numbers higher than 29, and all composite numbers who’s lowest prime factors are higher than 29,
are limited to the other 15.7947223% of the number line.
If we add the ratios for each prime number, one at a time and chart them, we can see that the
infinite series converges to the number 1. Since each ratio represents composite numbers per consecutive numbers, and it converges
to 1, this means that the ratio of prime numbers per all consecutive numbers is 0.
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