The Fifth Conjecture

Author : Spanu Dumitru Viorel Address: Street Marcu Mihaela Ruxandra no. 5 , 061524 , flat 47 , Bucharest , Romania E-mail : [email protected] [email protected] [email protected] Phones : +40214131107 0731522216

The Fifth Conjecture Let n be a natural even number , and that means n belongs to N \{ 0 } . Let i be a natural number and let be i

n = ------2

.

Then 1

∑ p p + n prime

where Ki

------p

are all of them

=

Ki

constants .

,

The sequence

1 ----pα

is repeated twice in each series ,

when the gap equal to n appears for the first time and then the sequence do not repeat in the frame of the series . Remark: When n is equal to 2 , then

K1 =

B

where B is the constant of Brun and it has the value B = 1,902160583104 …

.

In other words ( The Fifth Conjecture ) : The series [ in the plural ] made with the reciprocal of all the prime numbers which constitute pairs of prime numbers in which the distance is a gap equal to 2 , or a gap equal to 4 , …. , or a gap equal to n , are , all of these series [in the plural ] , convergent , and , each of these convergent series [in the plural ] is equal to a constant .

These constants represents the limits toward the above series [in the plural ] converge and all these constants are different one to another . These convergent series [in the plural ] which are made as it was described above are infinitely many .

The sequence

1 ----pα

is repeated twice in each series ,

when the gap equal to n appears for the first time and then the sequence does not repeat in the frame of the series .

Examples : 1/3 + 1/5 + 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/29 + 1/31 + 1/41 + 1/43 + 1/59 + 1/61 + 1/71 +1/73 … = B = The Brun`s Constant = 1,902160583104 … = K1 . 1/3 + 1/7 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/23 + 1/37 + 1/41 + 1/43 + 1/47 + 1/67 + 1/71 + … = K2 . 1/5 + 1/11 + 1/ 11 + 1/17 + 1/23 + 1/29 + 1/31 + 1/37 + 1/41 + 1/47 + 1/53 + 1/59 + 1/61 + 1/67 + … = K3 . 1/3 + 1/11 + 1/11 + 1/19 + 1/23 + 1/31 + 1/29 + 1/37 + 1/53 + 1/61 + 1/59 + 1/67 + 1/71 + 1/79 + … = K4 . 1/3 + 1/13 + 1/13 + 1/23 + 1/7 + 1/17 + 1/19 + 1/29 + 1/31 + 1/41 + 1/37 + 1/47 + 1/43 + 1/53 + 1/61 + 1/71 + 1/73 + 1/83 + 1/79 + 1/89 + … = K5 . 1/5 + 1/17 + 1/17 + 1/29 + 1/7 + 1/19 + 1/11 + 1/23 + 1/31 + 1/43 + 1/47 + 1/59 + 1/61 + 1/73 + 1/67 + 1/79 + 1/89 + 1/101 + … = K6 . ……………………………………………………………… 1/3 + 1/23 + 1/23 + 1/43 + 1/11 + 1/31 + 1/17 + 1/37 + 1/41 + 1/61 + 1/47 + 1/67 + 1/53 + 1/73 + 1/59 + 1/79 + … = K10 . ......................................................................................................