inteersting prime number observation

[phillip1882]

i'm not sure if this has been noticed before, but i descovered an iteresting prime fact. for any even number, there are an ifnite number of primes with that gap, and they occur within the range of the of the smaller prime to the smaller prime +the larger prime.
for example, lets take 8. the first prime +8 = prime, will be no more than 16.
3 +8 = 11.
the next prime +8 also prime, will be between 3 and 14
5 +8 = 13. next between 5 and 18.
11+8 = 19. 11-30
23+8 = 31

i've tested this with many even numbers and could not find an exception.

 

[HallsofIvy]

Taking the even number 2, " there are an infinite number of primes with that gap" would imply that there are an infinite number of "twin primes" and I don't believe that has ever been proven.

 

[phillip1882]

its never been proven, but i cant find a counter example. with the twin primes, my conjecture predicts the the next twin prime is less than 2* the previous twin prime

first twin prime < 4
3 5
next twin prime <8
5 7
next twin prime <12
11 13
next twin prime < 24
17 19
next twin prime <36
29 31 and so on.

 

[JeffM]

i'm not sure if this has been noticed before, but i descovered an iteresting prime fact. for any even number, there are an ifnite number of primes with that gap, and they occur within the range of the of the smaller prime to the smaller prime +the larger prime.
for example, lets take 8. the first prime +8 = prime, will be no more than 16.
3 +8 = 11.
the next prime +8 also prime, will be between 3 and 14
5 +8 = 13. next between 5 and 18.
11+8 = 19. 11-30
23+8 = 31

i've tested this with many even numbers and could not find an exception.

Quite frankly, I am not sure what you are trying to say. But if p > q > 2 and p and q are primes, then of course an even number divides p - q because both p and q are odd. That is a trivial result.

As Halls implied, it has been known for a long time that there are many pairs of primes such that p - q = 2, and there have been attempts to prove that the number of such pairs is infinite. A finite number of examples cannot prove a proposition about the infinite.

 

[phillip1882]

what i'm saying is that there are an infinite number of primes with any even gap, and given an even number, the next prime with that gap also prime will be less than 2*the larger previous prime. (i put a slightly more strict bound, but that's the general jist.)

 

[JeffM]

what i'm saying is that there are an infinite number of primes with any even gap

Fabulous. Where is your proof for 2?

the next prime with that gap also prime will be less than 2*the larger previous prime. (i put a slightly more strict bound, but that's the general jist.)

OK. That is clear, but again where is your proof?