what is the error in this prime gap counting logic?

The error is that this rubbish is continually posted in s.e.d., where it benefits precisely *no-one*. Please trim to sci.math only in future.

Reply to
Clifford Heath
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Hi,

I am trying to count the relative likelihood of gaps of 2,4 or 6 occurring in consecutive prime numbers, irrelevant of the overall absolute gaps in primes.

I am getting some differing results for what the expected likelihood of gaps 2,4,6 would be depending on the two sets of equations I am using.

This is also related to the twin prime conjecture to show it is always possible to have a gap of 2 in the primes if the prime formulas below create primes randomly.

equation set 1:

set of equations that produces all prime numbers (except 2,3,5):

y = 30x + 1 y = 30x + 7 y = 30x + 11 y = 30x + 13 y = 30x + 17 y = 30x + 19 y = 30x + 23 y = 30x + 29

For gaps of 2,4,6, that set of decoupled prime producing equations has the offsets:

1-7 11-7, 13-11, 17-13, 19-17, 23-19, 29-23, 29-1

which are prime gaps:

6, 4, 2, 4, 2, 4, 6, 2

(total of 8 prime gaps)

If the primes are randomly distributed there should always be a chance that there will be two primes for a given x, and then a 3/8 chance that the two x values give a prime gap of 2.

equation set 2:

set of equations that produces all prime numbers (except 2,3,5,7):

y= 210x + b

for b values:

1,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113, 121,127,131,137,139,143,149,151,157,163,167,169,173,179,181,187,191,193,197,199,209,

The list of gaps for the above is:

10,2,4,2,4,6,2,6,4,2,4,6,6,2,6,4,2,6,4,6,8,4,2,4,2,4,8,6,4,6,2,4,6,2,6,6,4,2,4,6,2,6,4,2,4,2,10

Total number of gaps is 47, and the count of gaps for 2,4,6 are:

gap count

2 14 4 15 6 14

Taking into account the gaps for 2,4,6 for x+1, ie just 209-1, there is one additional gap of 2 again to add, so it is 15 gaps of 2.

Now out of the possibilities of gaps 2,4,6 there is a 15/44 (0.3409) chance that the gap is 2.

This is a bit lower than for primorial 30 equation set 1 above, which had a 3/8 chance of the gap being 2, not sure why exactly?!?!

cheers, Jamie

Reply to
Jamie M

I think I figured it out, this just shows that the equation sets above aren't randomly distributed, ie it may be possible that there are NO gap of 2 consecutive equations from the above equation sets that are both prime if the twin prime conjecture is false.

Also even if the equation sets were randomly distributed, there is a bias away from gap=2 equations that can be seen in fewer gap of 2 possibilities in the equation sets. (ie from 3/8 to 15/44 reducing probability)

So in the two possible cases, either randomly distributed equations or non-randomly distributed equations yielding primes, the count of gap=2 primes might approach zero if larger primorial equation set blocks are checked for probability of gap=2 primes.

cheers, Jamie

Reply to
Jamie M

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