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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?

cheers
Ivan
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.
.
(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
(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.
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.
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