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 ?

@Vocaloid please

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

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

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

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

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?

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.

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"

what do you mean by "members sum" ?

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

@imqwerty now is more understandably ?

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?

yes - i think you get it right now are twin primes or members of different two twin primes

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

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

or 17 +19 = 36 36 + (5 -7) = 41 - 43

interesting problem

ty - any idea to prove it ?

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

\(\color{#0cbb34}{\text{Originally Posted by}}\) @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.

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`

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

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.

that's correct, my bad

ok. np

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 ?

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