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We start with the infinite set of all positive odd integers beginning with 3 (3,5,7,9…). Each group of two adjacent integers is a “candidate twin prime.”

Dividing each number in this infinite set by 3 (except 3 itself), we find that the number of remaining “candidate twin primes” between 3 and any higher odd number N is always at least 1/3 of the number previously in this subset. (Example: between 3 and 97, there were previously 47 “candidate twin primes” – now there are 17.) Repeating the process by dividing each number by 5 (except 5 itself), we now find that the number of remaining “candidate twin primes” between 3 and any higher odd number N is at least 1/5 of the original number in this subset. (Example: between 3 and 97, there were originally 47 “candidate twin primes” – now there are eleven. Division by 3 removed 30 candidates, and now division by 5 has removed a further 6 candidates.)

The same appears to hold true for all odd numbers up the series: Further division by 7 will result in a minimum of 1/7 of the original “candidate twin primes” remaining, division by 9 will result in result in a minimum of 1/9 of the original “candidate twin primes” remaining, etc. (Of course 9 is not prime and will not divide evenly into any of the remaining “candidate twin primes,” but that does not invalidate the above statement since a minimum of 1/9 is a subset of a minimum of 1/7.)

If this relationship holds true for all successive odd numbers N, we can show that the minimum number of twin primes between 3 and (N squared – 2) increases as the value of odd number N increases. For example, let N equal 7. All non-prime odd numbers less than 7 squared (3 through 47) are divisible by 3 or 5. Therefore, the number of actual twin primes between 3 and 47, inclusive, must be at least 1/5 of the original number of “candidate twin primes” in this subset. As the number of “candidate twin primes” was initially 22, there must be at least 4.4 (rounded up to 5) twin primes below 7 squared. (In fact there are 6.)

Now let N equal 9. All non-prime odd numbers less than 9 squared (3 through 79) are divisible by 3, 5 or 7. Therefore, the number of twin primes between 3 and 79, inclusive, must be at least 1/7 of the original number of “candidate twin primes” in this subset. As the number of “candidate twin primes” was initially 38, there must be at least 5.42… (rounded up to 6) twin primes below 9 squared. (In fact there are 8.)

Successive iterations of this procedure show that for each increase in N to the next higher odd number, the minimum number of twin primes between 3 and (N squared – 2) increases by more than 1.

Since there are an infinite number of possible values of N (an infinite number of positive odd integers 3 or greater), and each of these values is associated with a discrete positive integer value for the minimum number of twin primes between 3 and (N squared – 2), there must therefore be an infinite number of twin primes.

However, this result depends crucially upon whether the initial premise above is true – that is, whether successive divisions of all positive odd integers by all odd integers 3 through N always leave at least 1/N of the original number of “twin prime candidates” between 3 and any higher odd number.

Has this premise or a similar one been proven or disproven? If not, would there be a straightforward way to do so?