Prime number + 2 = Prime number

[Ivan Rancati]

EDIT: oops. Twin primes, plenty of study on that already. Sorry for the old news


I read this riddle in a 1950s encyclopedia (in Italian), and it piqued my interest.

Adding some prime numbers to 2 still yields a prime number
For example
3, 5
5, 7
11, 13
17, 19
41, 43
and so on

How could one possibly go about proving that there is an infinite number of these "prime pairs", or that past a certain prime number there is never going to be such a pair?


I would imagine there is already some conjecture about this, but not sure

cheers
Ivan

[Ron Ross]

I was going to suggest looking at Mersenne prime numbers, as they can be used to help find these pair of primes separated by 2.
.

[Vtile]

(Any integer above number 2) = 2 * A + 3 * B (in my mind)

5 = 2 * 1 + 3 * 1
7 = 2 * 2 + 3 * 1
8 = 2 * 4 + 3 * 0
9 = 2 * 0 + 3 * 3
10 = 2 * 2 + 3 * 2
11 = 2 * 4 + 3 * 1
12 = 2 * 6 + 3 * 0
13 = 2 * 2 + 3 * 3
14 = 2 * 7 + 3 * 0
15 = 2 * 6 + 3 * 1
16 = 2 * 2 + 3 * 4
17 = 2 * 7 + 3 * 2
18 = 2 * 6 + 3 * 2
19 = 2 * 8 + 3 * 1
...
41 = 2 * 19 + 3 * 1
43 = 2 * 19 + 3 * 3
as seen there is plenty of possibilities that that prime + 2 is prime...

Soo there is also prime + 4 = primes (37, 41) and prime + 3 = prime(s?) (2, 5) and prime + 6 = primes etc..

In that light this weren't a suprise for me, although I see it as interesting pattern
that I'm not interested to search, but like..

Code:


 1   ? 0?
 2 , 1 
 3,  1
 5,  2
 7,  2
 11, 4 
 13, 2 
 17, 4 
 19, 2
 23, 4
 29, 6 <- 1st 6 = 2*3 or 2^2 + 3^0 or 2^2 + 2^0
 31, 2
 37, 6
 41, 4
 43, 2
 47, 4
 53, 6
 59, 6
 61, 2
 67, 6
 71, 4
 73  2

So 1001 can be written in as 500d, 1b as 2*500 + 2^0 or something like 250d,11b yeah, I'm inventing as writing so this makes no sense per se so time to go to sleep.
edit2 typo 2^1 -> 2^0

[John R]

(06-24-2016 07:53 PM)Ivan Rancati Wrote:  How could one possibly go about proving that there is an infinite number of these "prime pairs", or that past a certain prime number there is never going to be such a pair?

There have been some recent developments in this regard.

[Vtile]

Thx for the OP for this thread, made me 1st time (since educational math is as interesting as wet socks ) to think about nature of the complementary things in maths.