Counting the prime pairs


Is there an infinite number of prime pairs?

How frequent are prime pairs?

Why are their prime pairs?

Intriguing questions best examined from the composites: why do they exist for starters? Look at the role of prime factors 2 and 3. They lead to the first prime pair 5&7.

Look at the role of prime factors 2, 3 and 5. 41/43 and 59/61 show up, but 37/39 and 53/55 would be pairs without the effects of 13 and 11.

We can show the number of pairs per length of composite numbers thus:

It starts:

PF 2,3: 1 pair for each 5 composites (20% or 40% of composites are one of a pair)

PF 2,3,5: 4 pairs for 30 composites (13.3% or 26.6% of composites are one of a pair)

PF 2,3,5,7: 19 pairs in 210 composites (9.0% or 18.1% of composites are one of a pair)

and so on. Can we establish a formula and work out whether it has an asymmtote?

(p-1)#/p#

 

 

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