Hi math lovers ! please opinions about this : than i can prove that always there exist a prim number the form of 6n+/-1 what adding to the members sum of a twin primes always result - generate - a new twin prime or two different twin prime members - so and my question about this : maybe this proof prove the infinity property of the twin primes line ?

Question

jhonyy9 · Answer

@Vocaloid please

darkknight · Answer

hmm... maybe @imqwerty ? Could you try this?

jhonyy9 · Answer

@Hero please any idea about this ? thank you in advance

Timmyspu · Answer

yes u can prove this it would take u a while tho

jhonyy9 · Answer

so I proved it, but it depends on whether it will be accepted or not

imqwerty · Answer

so you want to prove that there always exist a prime number of form 6n$\pm$1 for every twin primes, such that on adding the prime number to each of the twin primes you get two new twin primes?

imqwerty · Answer

consider the twin primes as $p_1$ and $p_2$  and consider any prime number $p$

will there exist a $p$ such that $p_1 + p$ and $p_2 + p$ are twin primes

welp, all the prime numbers except for 2 are odd, odd + odd = even.

considering the case for twin primes ($p_1$ & $p_2$): 5 & 7
there exists no prime ( $p$ ) such that (5 + p) & (7 + p) are twin primes, because both the sum are even. Or if p=2, the generated numbers are - 7 & 9 which aren't twin primes.

jhonyy9 · Answer

ATTENTION please - i ve wrote in a different way in my posted text

,, what adding to the members sum of a twin primes always result - generate - a new twin prime or two different twin prime members"

imqwerty · Answer

what do you mean by "members sum" ?

jhonyy9 · Answer

ok - for example

twin prime the form of 6n +/- 1 let be the first one of 5-7 so 5+7 = 12

always there exist a twin prime the form of 6n +/- 1 what assuming to 12 what is the sum of the twin prime 5-7 so that generate a new twin prime or new twin primes members

12 + (5 - 7) = 17 - 19 
or

12 + (29 - 31) = 41 - 43

jhonyy9 · Answer

@imqwerty now is more understandably ?

imqwerty · Answer

i see, this is what i've understood:

Prove that there always exist twin primes $\left(p_1, p_2\right)$ of the form $6n\pm1$  for each twin primes $\left(p_3, p_4\right)$ such that $\left(p_1 + p_2 + p_3, p_1+p_2+p_4\right)$ are twin primes.

Did i get it right?

jhonyy9 · Answer

yes - i think you get it right now

are twin primes or members of different two twin primes

imqwerty · Answer

i think a part of this proof would also require us to prove that there are infinitely many twin primes of the form $\left(6n-1, 6n+1\right)$

jhonyy9 · Answer

yes i understand what you say but look please in this way we start with the first one

5 - 7 so 5+7 = 12

12 + (5 - 7 ) = 17 - 19 and in this way this is a new twin prime the form of 6n +/- 1

and we know the second twin the form of 6n +/- 1 is 11-13

11 +13 = 24

24 + (5 - 7 ) = 29 - 31 what is a new twin prime the form of 6n +/ - 1

jhonyy9 · Answer

or 17 +19 = 36

36 + (5 -7) = 41 - 43

imqwerty · Answer

interesting problem

jhonyy9 · Answer

ty - any idea to prove it ?

jhonyy9 · Answer

so this maybe my conjecture - ipoteza - what i ve ,,get" - edited - this year in September

imqwerty · Answer

$\color{#0cbb34}{\text{Originally Posted by}}$ @imqwerty
i think a part of this proof would also require us to prove that there are infinitely many twin primes of the form $\left(6n-1, 6n+1\right)$
$\color{#0cbb34}{\text{End of Quote}}$
i'm feeling like prooving this might help us move forward. 
Not all $\left(6n \pm 1\right)$ are twin primes. So we would have to figure out the $n$ values which make it a twin-prime pair. 
To find those values, i guess we could run a program to find some n-values for us so that we could find a pattern in those values.

jimthompson5910 · Answer

All primes are of the form $6m\pm 1$ where m is an integer https://math.stackexchange.com/questions/41623/is-that-true-that-all-the-prime-numbers-are-of-the-form-6m-pm-1 Though be careful and note that m = 6 leads to 6*6-1 = 36-1 = 35 = 5*7 which is not prime. This is one example of many. So the statement should be carefully crafted and say "If a number is prime, then it is in the form 6m-1 or 6m+1 for some integer m". The reverse statement "if a number is in the form 6m-1 or 6m+1, for any integer m, then the number is prime" is false. Saying "twin primes of the form 6m-1 and 6m+1" is a bit redundant in my opinion. All twin primes fit this form. Also, I agree with imqwerty that you would need to prove there exists infinitely many twin primes. Currently, it's not known if there are infinitely many or not https://en.wikipedia.org/wiki/Twin_prime That page says `However, it is unknown whether there are infinitely many twin primes or there is a largest pair`

jhonyy9 · Answer

yes - i agree with you above wrote but : Saying "twin primes of the form 6m-1 and 6m+1" is a bit redundant in my opinion. All twin primes fit this form.

- so this is not exactly true bc. 3-5 is accepted the first twin prime

jimthompson5910 · Answer

Sorry I should clarify and say the twin pair (3,5) is not of the form $6m\pm1$ but every other twin prime pair does fit the description.

jimthompson5910 · Answer

that's correct, my bad

jhonyy9 · Answer

ok. np

jhonyy9 · Answer

so my new idea to edit this posted problem in this way

... if we accept that the line of twin primes the form of 6n +/- 1 is infinitely so then ... and my posted problem

i think in this way maybe true - any idea now ?