Relative twins frequency
- Hi all,
Consider the set of natural numbers and filtering them by 3,5 and 7 (but not 2!)
Within an interval of 105 I will find 15 pairs of numbers that differ by 2.
If I add 11 to the filtering set and look at the interval which of course is the product of the filtering set i.e. 18.104.22.168 = 1155. I will find 3.5.9 = 135 pairs which differ by two.
I want help in considering the idea of the number of pairs I might expect in fractions of the product of the filtering primes.
I understand that the distribution of the pairs is 'irregular' and the largest gap between two pairs is perhaps unknown for any n, number of filtering primes.
I can see that given the paragraph above one cannot simply say that since there are 15 pairs in 105 therefore there will be 15/3 pairs in 105/3 or that since there are 135 pairs within the interval 1155 then there will be 135/5 pairs within an interval of 1155/5 etc.
But surely there is some kind of generalisation that one can use for fractions of these intervals.
Can anyone lead me to an understanding of whether there is such a generalised process that can be applied and what it is?
Just in case I have been a bit vague at the end there; I might expand my request by saying that with the interval 1155; if I wanted to consider how many pairs I might expect within the fraction 1155/15; I might say something like there are 77 + or - a; pairs in that fraction; where a is a whole number. Where 'a' could denote the degree of inaccuracy of course.
Have I asked a silly question here? If so, then could someone enlighten me as to why?
Thankyou in advance for any consideration.