Mathematics jhonyy9:

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 ? jhonyy9: darkknight:

hmm... maybe @imqwerty ? Could you try this? jhonyy9: Timmyspu:

yes u can prove this it would take u a while tho jhonyy9:

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

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:

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:

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:

what do you mean by "members sum" ? jhonyy9:

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:

@imqwerty now is more understandably ? imqwerty:

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:

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

i think a part of this proof would also require us to prove that there are infinitely many twin primes of the form $$\left(6n-1, 6n+1\right)$$ jhonyy9:

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:

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

interesting problem jhonyy9:

ty - any idea to prove it ? jhonyy9:

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

$$\color{#0cbb34}{\text{Originally Posted by}}$$ @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:

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:

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:

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: jhonyy9:

ok. np jhonyy9:

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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